# Given Information Events A and B are independent P(A) = .4 P(A ∪ B) = .7 Required: Find P(B) # Definitions and Concepts Used - If A and B are independent, then P(A ∩ B) = P(A) × P(B). - The probability of the union of two events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) # Solution Write the formula for the union: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) Since A and B are independent: P(A ∩ B) = P(A) × P(B) Substitute the given values and the above relation into the union formula: .7 = .4 + P(B) − [.4 × P(B)] Express .4 × P(B) as .4P(B): .7 = .4 + P(B) − .4P(B) Combine like terms: .7 = .4 + [P(B) − .4P(B)] .7 = .4 + [1 − .4]P(B) .7 = .4 + .6P(B) Subtract .4 from both sides: .7 − .4 = .6P(B) .3 = .6P(B) Divide both sides by .6: P(B) = .3 / .6 P(B) = .5 So, the probability of event B is .5.

# Given Information - Events A and B are independent - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Independence of Events**: For independent events A and B, \[ P(A \cap B) = P(A) \times P(B) \] - **Probability of Union**: The probability of the union of two events is given by \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] # Solution Start by applying the formula for the union of events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Given that A and B are independent, substitute for \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Now, substitute the known values into the equation: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rewrite \( 0.4 \times P(B) \) as \( 0.4P(B) \): \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine the terms involving \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] Next, isolate \( P(B) \) by subtracting \( 0.4 \) from both sides: \[ 0.7 - 0.4 = 0.6P(B) \] This leads to: \[ 0.3 = 0.6P(B) \] To find \( P(B) \), divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \( P(B) = 0.5 \).

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Independence of Events**: For independent events A and B, the probability of their intersection is given by \[ P(A \cap B) = P(A) \times P(B) \] - **Probability of Union**: The probability of the union of two events is calculated as \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] # Solution Start with the probability of the union formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Given the independence of A and B, substitute \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Substitute the known values into the equation: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rewrite \( 0.4 \times P(B) \) as \( 0.4P(B) \): \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Group the terms involving \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This results in: \[ 0.7 = 0.4 + 0.6P(B) \] Now, isolate \( P(B) \) by subtracting \( 0.4 \) from both sides: \[ 0.7 - 0.4 = 0.6P(B) \] This simplifies to: \[ 0.3 = 0.6P(B) \] To determine \( P(B) \), divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Therefore, the probability of event B is \( P(B) = 0.5 \).

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Independence of Events**: For independent events A and B, the probability of their intersection is expressed as \[ P(A \cap B) = P(A) \times P(B) \] - **Probability of Union**: The probability of the union of two events is given by the formula \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] # Solution Begin with the formula for the probability of the union of events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since A and B are independent, replace \( P(A \cap B) \) with its equivalent: \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Insert the known values into the equation: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Express \( 0.4 \times P(B) \) as \( 0.4P(B) \): \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Reorganize the terms involving \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] Next, isolate \( P(B) \) by subtracting \( 0.4 \) from both sides: \[ 0.7 - 0.4 = 0.6P(B) \] This results in: \[ 0.3 = 0.6P(B) \] To find \( P(B) \), divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \( P(B) = 0.5 \).

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Independence of Events**: For independent events A and B, the intersection probability is calculated as \[ P(A \cap B) = P(A) \times P(B) \] - **Probability of Union**: The union of two events A and B is given by the formula \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] # Solution Start by applying the union formula for events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since A and B are independent, substitute \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Substitute the known values into the equation: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging \( 0.4 \times P(B) \) gives: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine the terms involving \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] Next, isolate \( P(B) \) by subtracting \( 0.4 \) from both sides: \[ 0.7 - 0.4 = 0.6P(B) \] This leads to: \[ 0.3 = 0.6P(B) \] To find \( P(B) \), divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Therefore, the probability of event B is \( P(B) = 0.5 \).

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Independence of Events**: If A and B are independent, then \[ P(A \cap B) = P(A) \times P(B) \] - **Probability of Union**: The probability of the union of events A and B is defined as \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] # Solution Begin by applying the formula for the union of events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substituting the expression for \( P(A \cap B) \) based on independence: \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Insert the known values into this equation: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging \( 0.4 \times P(B) \): \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine terms that include \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), subtract \( 0.4 \) from both sides: \[ 0.7 - 0.4 = 0.6P(B) \] This results in: \[ 0.3 = 0.6P(B) \] To find \( P(B) \), divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \( P(B) = 0.5 \).

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Independence of Events**: For independent events A and B, the intersection probability can be calculated as \[ P(A \cap B) = P(A) \times P(B) \] - **Probability of Union**: The formula for the union of two events A and B is \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] # Solution Start with the union formula for events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Given that A and B are independent, substitute \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Now plug in the known values: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging gives: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine the terms involving \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), subtract \( 0.4 \) from both sides: \[ 0.7 - 0.4 = 0.6P(B) \] This results in: \[ 0.3 = 0.6P(B) \] Now, divide both sides by \( 0.6 \) to find \( P(B) \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \( P(B) = 0.5 \).

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Event Independence**: Two events A and B are independent if the occurrence of one does not affect the other. Mathematically, this is represented as \[ P(A \cap B) = P(A) \times P(B) \] where \( P(A \cap B) \) is the probability that both events occur. - **Probability of Union**: The probability that at least one of the events A or B occurs is given by the formula \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] where \( P(A \cup B) \) is the probability that either A or B occurs. # Solution To find \( P(B) \), we start with the formula for the union of events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Given that A and B are independent, we can substitute \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Now substitute the known values: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging \( 0.4 \times P(B) \): \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine the terms that contain \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] Next, isolate \( P(B) \) by subtracting \( 0.4 \) from both sides: \[ 0.7 - 0.4 = 0.6P(B) \] This results in: \[ 0.3 = 0.6P(B) \] To find \( P(B) \), divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \]

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Independence of Events**: Events A and B are independent if the occurrence of one does not influence the occurrence of the other. This is mathematically expressed as: \[ P(A \cap B) = P(A) \times P(B) \] where \( P(A \cap B) \) is the probability that both events A and B occur simultaneously. - **Probability of Union**: The probability that at least one of the events A or B occurs is given by the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] where \( P(A \cup B) \) is the probability that either event A or event B occurs. # Solution Start with the union formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since A and B are independent, we can substitute \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Substituting the known values into this equation gives: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging \( 0.4 \times P(B) \): \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine the terms involving \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), subtract \( 0.4 \) from both sides: \[ 0.7 - 0.4 = 0.6P(B) \] This results in: \[ 0.3 = 0.6P(B) \] Now, divide both sides by \( 0.6 \) to find \( P(B) \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \]

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Independence of Events**: For independent events A and B, \[ P(A \cap B) = P(A) \times P(B) \] - **Probability of Union**: The probability of the union of events A and B is given by \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] # Solution Use the union formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substituting the expression for \( P(A \cap B) \) based on independence: \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Now, substitute the known values: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging gives: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine terms involving \( P(B) \): \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), subtract \( 0.4 \): \[ 0.3 = 0.6P(B) \] Dividing both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \]

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Independence of Events**: Two events A and B are independent if the occurrence of one does not affect the other. This is mathematically defined by: \[ P(A \cap B) = P(A) \times P(B) \] - **Probability of Union**: The probability that at least one of the events A or B occurs is calculated using: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] # Solution Start with the union formula for events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Given that A and B are independent, we can express \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Now, substituting the given values: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging gives: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine the terms involving \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), subtract \( 0.4 \): \[ 0.3 = 0.6P(B) \] Now, divide by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \]

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Independence of Events**: For independent events A and B, the probability of both occurring is calculated as: \[ P(A \cap B) = P(A) \times P(B) \] - **Probability of Union**: The probability that at least one of the events occurs is given by the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] # Solution Start by applying the union formula for events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since A and B are independent, substitute \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Substituting the known values into the equation: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging the equation gives: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine terms related to \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] Next, isolate \( P(B) \) by subtracting \( 0.4 \) from both sides: \[ 0.3 = 0.6P(B) \] To find \( P(B) \), divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \]

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Independence of Events**: Events A and B are independent if the occurrence of one does not influence the occurrence of the other. This is mathematically represented as: \[ P(A \cap B) = P(A) \times P(B) \] - **Probability of Union**: The probability that at least one of the events A or B occurs is given by the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] # Solution Start with the union formula for events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Given the independence of A and B, substitute \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Now substitute the known values into the equation: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging gives: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine the terms involving \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), subtract \( 0.4 \) from both sides: \[ 0.3 = 0.6P(B) \] Now, divide by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \]

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Independence of Events**: Two events A and B are independent if the occurrence of one does not affect the probability of the other. This is defined mathematically as: \[ P(A \cap B) = P(A) \times P(B) \] - **Probability of Union**: The probability that at least one of the events A or B occurs can be expressed using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] # Solution Start by applying the union formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since A and B are independent, we can replace \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Now substitute the known values into the equation: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging gives: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine the terms related to \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] Next, isolate \( P(B) \) by subtracting \( 0.4 \): \[ 0.3 = 0.6P(B) \] To find \( P(B) \), divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \]

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Independence of Events**: Two events A and B are independent if the occurrence of one does not affect the probability of the other. This is mathematically defined as: \[ P(A \cap B) = P(A) \times P(B) \] - **Probability of Union**: The probability that at least one of the events A or B occurs is expressed by the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] # Solution Begin with the union formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since A and B are independent, we substitute \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Now, substitute the known values: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging gives: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine the terms involving \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] Isolate \( P(B) \) by subtracting \( 0.4 \): \[ 0.3 = 0.6P(B) \] To determine \( P(B) \), divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \]

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Required: Find \( P(B) \) # Definitions and Concepts Used - **Independence of Events**: Two events A and B are considered independent when the occurrence of one does not influence the occurrence of the other. Mathematically, this relationship is expressed as: \[ P(A \cap B) = P(A) \times P(B) \] Here, \( P(A \cap B) \) represents the probability that both events A and B occur simultaneously. - **Probability of Union**: The probability that at least one of the events A or B occurs can be calculated using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] In this formula, \( P(A \cup B) \) denotes the probability that either event A occurs, event B occurs, or both events occur. # Solution To find \( P(B) \), we will begin with the union formula for events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since A and B are independent, we substitute \( P(A \cap B) \) with its equivalent expression: \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Next, we insert the known values into this equation: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging the equation allows us to isolate terms more clearly: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Next, we group the terms that involve \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] Now, to isolate \( P(B) \), we first subtract \( 0.4 \) from both sides of the equation: \[ 0.7 - 0.4 = 0.6P(B) \] This calculation gives us: \[ 0.3 = 0.6P(B) \] To find \( P(B) \), we will divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, we conclude that the probability of event B is \[ P(B) = 0.5 \] This result indicates that there is a 50% chance that event B will occur, given the independent nature of events A and B and the provided probabilities.

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - The task is to determine \( P(B) \). # Definitions and Concepts Used - **Independence of Events**: Two events A and B are independent when the occurrence of one does not affect the occurrence of the other. This can be mathematically expressed as: \[ P(A \cap B) = P(A) \times P(B) \] Here, \( P(A \cap B) \) represents the probability that both events happen simultaneously. - **Probability of Union**: The probability that at least one of the events A or B occurs can be calculated using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] In this context, \( P(A \cup B) \) represents the probability of either event A occurring, event B occurring, or both occurring. # Solution To find \( P(B) \), we will start with the formula for the union of events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Given that A and B are independent, we can rewrite the intersection probability: \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Substituting the known values into this equation provides: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging the equation helps to clarify the terms: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Next, we combine the terms that involve \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), we subtract \( 0.4 \) from both sides: \[ 0.7 - 0.4 = 0.6P(B) \] This leads to: \[ 0.3 = 0.6P(B) \] To solve for \( P(B) \), divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \] This outcome indicates that there is a 50% likelihood of event B occurring, considering the independence of events A and B along with the given probabilities.

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Task: Find \( P(B) \). # Definitions and Concepts Used - **Independence of Events**: Events A and B are independent if the occurrence of one does not affect the probability of the other. This is mathematically represented as: \[ P(A \cap B) = P(A) \times P(B) \] - **Probability of Union**: The probability that either event A or event B (or both) occurs can be expressed as: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Here, \( P(A \cup B) \) signifies the probability that at least one of the events occurs. # Solution To find \( P(B) \), we start with the union formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since A and B are independent, we can substitute \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Now, inserting the known values into this equation yields: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging gives: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine the terms involving \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), subtract \( 0.4 \): \[ 0.3 = 0.6P(B) \] Now, divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \] This means there is a 50% chance that event B will occur, given the independence of events A and B and the probabilities provided.

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Objective: Determine \( P(B) \). # Definitions and Concepts Used - **Independence of Events**: Two events A and B are independent if the occurrence of one does not influence the occurrence of the other. This relationship is expressed mathematically as: \[ P(A \cap B) = P(A) \times P(B) \] Here, \( P(A \cap B) \) denotes the probability that both events A and B occur together. - **Probability of Union**: The probability that at least one of the events A or B occurs can be calculated using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] In this formula, \( P(A \cup B) \) represents the probability of either event A occurring, event B occurring, or both. # Solution To find \( P(B) \), we will begin with the union formula for events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since A and B are independent, we can replace \( P(A \cap B) \) with: \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Inserting the known values into this equation gives: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging the terms allows for clearer isolation: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine the terms that involve \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), subtract \( 0.4 \) from both sides: \[ 0.3 = 0.6P(B) \] Now, divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \] This result indicates that there is a 50% chance that event B will occur, based on the independence of events A and B and the specified probabilities.

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Task: Calculate \( P(B) \). # Definitions and Concepts Used - **Independence of Events**: Events A and B are independent if the occurrence of one does not influence the occurrence of the other. This can be mathematically represented as: \[ P(A \cap B) = P(A) \times P(B) \] Here, \( P(A \cap B) \) represents the probability that both events A and B happen at the same time. - **Probability of Union**: The probability that at least one of the events A or B occurs is defined by the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] In this context, \( P(A \cup B) \) signifies the likelihood of either event A occurring, event B occurring, or both events occurring. # Solution To find \( P(B) \), we initiate with the union formula for events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since A and B are independent, we can substitute \( P(A \cap B) \) with: \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Substituting the known values into this equation results in: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging helps isolate the terms involving \( P(B) \): \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Next, we combine the terms that include \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), we first subtract \( 0.4 \) from both sides: \[ 0.3 = 0.6P(B) \] Now, divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \] This indicates that there is a 50% chance of event B occurring, given the independence of events A and B along with the provided probabilities.

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Objective: Determine \( P(B) \). # Definitions and Concepts Used - **Independence of Events**: Two events A and B are considered independent if the occurrence of one does not affect the probability of the other. This relationship is defined mathematically as: \[ P(A \cap B) = P(A) \times P(B) \] Here, \( P(A \cap B) \) represents the probability that both events A and B occur simultaneously. - **Probability of Union**: The probability that at least one of the events A or B occurs can be calculated using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] In this formula, \( P(A \cup B) \) represents the probability of either event A, event B, or both occurring. # Solution To find \( P(B) \), we will start with the union formula for events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since A and B are independent, we can rewrite \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Next, we substitute the known values into this equation: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging gives: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine the terms that involve \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), we first subtract \( 0.4 \): \[ 0.3 = 0.6P(B) \] Now, divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \] This indicates that there is a 50% chance that event B will occur, considering the independence of events A and B and the provided probabilities.

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Task: Find \( P(B) \). # Definitions and Concepts Used - **Independence of Events**: Two events A and B are independent if the occurrence of one does not affect the occurrence of the other. This is expressed mathematically as: \[ P(A \cap B) = P(A) \times P(B) \] Here, \( P(A \cap B) \) denotes the probability that both events A and B occur. - **Probability of Union**: The probability that at least one of the events A or B occurs can be calculated using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] In this context, \( P(A \cup B) \) signifies the probability of either event A occurring, event B occurring, or both. # Solution To determine \( P(B) \), we start with the union formula for events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since A and B are independent, we can substitute \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Substituting the known values into this equation yields: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging helps to isolate the terms: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Next, we group the terms involving \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), subtract \( 0.4 \) from both sides: \[ 0.3 = 0.6P(B) \] Now, dividing both sides by \( 0.6 \) gives: \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \] This outcome indicates that there is a 50% chance of event B occurring, considering the independence of events A and B along with the provided probabilities.

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Objective: Determine \( P(B) \). # Definitions and Concepts Used - **Independence of Events**: Two events A and B are independent if the occurrence of one does not influence the occurrence of the other. This is mathematically represented as: \[ P(A \cap B) = P(A) \times P(B) \] Where \( P(A \cap B) \) denotes the probability that both events A and B occur at the same time. - **Probability of Union**: The probability that at least one of the events A or B occurs is calculated using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] In this formula, \( P(A \cup B) \) signifies the likelihood of either event A or event B (or both) occurring. # Solution To determine \( P(B) \), we will start with the formula for the union of events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Given the independence of A and B, we substitute \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Now substituting the given values into this equation yields: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging helps isolate the terms involving \( P(B) \): \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Next, we combine the terms that involve \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), we will subtract \( 0.4 \) from both sides: \[ 0.3 = 0.6P(B) \] Now, dividing both sides by \( 0.6 \) gives: \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \] This indicates that there is a 50% chance that event B will occur, given the independence of events A and B and the specified probabilities.

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Goal: Calculate \( P(B) \). # Definitions and Concepts Used - **Independence of Events**: Events A and B are termed independent if the occurrence of one does not influence the occurrence of the other. This is quantitatively expressed as: \[ P(A \cap B) = P(A) \times P(B) \] Here, \( P(A \cap B) \) indicates the probability that both events happen simultaneously. - **Probability of Union**: The probability that at least one of the events A or B occurs is calculated using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] In this expression, \( P(A \cup B) \) represents the likelihood of event A occurring, event B occurring, or both events occurring. # Solution To find \( P(B) \), we will begin with the union formula for events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since A and B are independent, we can replace \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Next, substitute the known values into this equation: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging this equation allows us to group like terms: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] This can be simplified to: \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This further reduces to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), subtract \( 0.4 \) from both sides: \[ 0.7 - 0.4 = 0.6P(B) \] This results in: \[ 0.3 = 0.6P(B) \] Now, divide both sides by \( 0.6 \) to solve for \( P(B) \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \] This indicates that there is a 50% probability that event B will occur, given the independence of events A and B and the stated probabilities. This analysis illustrates the fundamental principles of probability theory, particularly the concepts of independence and the calculation of the union of two events.

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Task: Determine \( P(B) \). # Definitions and Concepts Used - **Independence of Events**: Events A and B are independent if the outcome of one event does not affect the outcome of the other. This relationship can be expressed mathematically as: \[ P(A \cap B) = P(A) \times P(B) \] Here, \( P(A \cap B) \) represents the probability that both events A and B occur simultaneously. - **Probability of Union**: The probability that at least one of the events A or B occurs is calculated using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] In this formula, \( P(A \cup B) \) signifies the likelihood of either event A, event B, or both events occurring. # Solution To find \( P(B) \), we start with the union formula for events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Since A and B are independent, we can express \( P(A \cap B) \) as: \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Now, substituting the known values into this equation yields: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging the equation helps in isolating the terms: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Combine terms involving \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), subtract \( 0.4 \) from both sides: \[ 0.3 = 0.6P(B) \] Now, divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \] This means there is a 50% chance that event B will occur, based on the independence of events A and B and the provided probabilities. This solution illustrates the application of basic probability principles, focusing on the concepts of independence and union of events.

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Goal: Calculate \( P(B) \). # Definitions and Concepts Used - **Independence of Events**: Two events A and B are considered independent when the occurrence of one does not influence the occurrence of the other. This can be mathematically expressed as: \[ P(A \cap B) = P(A) \times P(B) \] Here, \( P(A \cap B) \) represents the probability that both events A and B occur simultaneously. - **Probability of Union**: The probability that at least one of the events A or B occurs can be calculated using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] In this expression, \( P(A \cup B) \) signifies the probability of either event A occurring, event B occurring, or both events occurring. # Solution To find \( P(B) \), we will begin with the union formula for events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Given that A and B are independent, we can substitute \( P(A \cap B) \): \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Next, we substitute the known values into this equation: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging the equation helps isolate terms: \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] Now, combine the terms that involve \( P(B) \): \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] To isolate \( P(B) \), we subtract \( 0.4 \): \[ 0.3 = 0.6P(B) \] Now, dividing both sides by \( 0.6 \) gives: \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \] This means that there is a 50% chance that event B will occur, given the independence of events A and B along with the stated probabilities. This solution highlights the fundamental principles of probability, emphasizing the significance of independence and the calculation of the union of events.

# Given Information - Events A and B are independent. - \( P(A) = 0.4 \) - \( P(A \cup B) = 0.7 \) - Objective: Calculate \( P(B) \). # Definitions and Concepts Used - **Independence of Events**: Two events A and B are independent when the occurrence of one does not affect the occurrence of the other. This mathematical relationship can be defined as: \[ P(A \cap B) = P(A) \times P(B) \] Here, \( P(A \cap B) \) represents the probability that both events A and B occur simultaneously. - **Probability of Union**: The probability that at least one of the events A or B occurs is calculated using the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] In this expression, \( P(A \cup B) \) denotes the combined probability of event A occurring, event B occurring, or both events occurring. # Solution To determine \( P(B) \), we start with the union formula for events A and B: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Given the independence of A and B, we can express \( P(A \cap B) \) as: \[ P(A \cup B) = P(A) + P(B) - (P(A) \times P(B)) \] Next, we substitute the known values into this equation: \[ 0.7 = 0.4 + P(B) - (0.4 \times P(B)) \] Rearranging the equation helps isolate the terms involving \( P(B) \): \[ 0.7 = 0.4 + P(B) - 0.4P(B) \] By grouping the terms related to \( P(B) \), we have: \[ 0.7 = 0.4 + (1 - 0.4)P(B) \] This simplifies to: \[ 0.7 = 0.4 + 0.6P(B) \] Next, to isolate \( P(B) \), we subtract \( 0.4 \) from both sides: \[ 0.3 = 0.6P(B) \] To find \( P(B) \), we divide both sides by \( 0.6 \): \[ P(B) = \frac{0.3}{0.6} = 0.5 \] Thus, the probability of event B is \[ P(B) = 0.5 \] This indicates that there is a 50% chance that event B will occur. This result is particularly significant as it reflects the independence of events A and B; hence, the occurrence of one does not impact the likelihood of the other. Understanding these probability concepts is crucial in various fields such as statistics, finance, and decision-making, where assessing the likelihood of multiple events is essential.