Let's answer part (a): ### Step-by-Step Solution #### **Step 1: Find the Total Number of Individuals** Sum all the values in the table: - **Never smoked cigars:** \( 583 + 119,523 = 120,106 \) - **Former cigar smoker:** \( 91 + 7,124 = 7,215 \) - **Current cigar smoker:** \( 163 + 6,373 = 6,536 \) **Total:** \( 120,106 + 7,215 + 6,536 = 133,857 \) #### **Step 2: Find the Number Who Died from Cancer** Sum the "Died from Cancer" column: - Never smoked cigars: 583 - Former cigar smoker: 91 - Current cigar smoker: 163 **Total who died from cancer:** \( 583 + 91 + 163 = 837 \) #### **Step 3: Calculate the Probability** \[ P(\text{died from cancer}) = \frac{\text{Number who died from cancer}}{\text{Total number of individuals}} \] \[ P(\text{died from cancer}) = \frac{837}{133,857} \approx 0.006 \] ### **Final Answer (rounded to three decimal places):** \[ \boxed{0.006} \]

## Data Overview The study involves 133,857 men, examining the association between cigar smoking and death from cancer. The table provides counts of individuals who died from cancer and those who did not, segmented by smoking status. ### Data Table Summary | Smoking Status | Died from Cancer | Did Not Die from Cancer | |-------------------------|------------------|-------------------------| | Never smoked cigars | 583 | 110,523 | | Former cigar smoker | 51 | 7,124 | | Current cigar smoker | 163 | 6,373 | ### Total Counts - **Total individuals:** 133,857 - **Total who died from cancer:** 837 --- ## Solution to Parts (a) - (d) ### (a) Probability that an individual died from cancer \[ P(\text{died from cancer}) = \frac{\text{Number who died from cancer}}{\text{Total number of individuals}} = \frac{837}{133,857} \approx 0.006 \] ### **Final Answer (a):** \[ \boxed{0.006} \] --- ### (b) Probability that an individual was a current cigar smoker #### **Step 1: Count current cigar smokers** Total current cigar smokers = 163 (died) + 6,373 (did not die) = 6,536 #### **Step 2: Calculate probability** \[ P(\text{current cigar smoker}) = \frac{6,536}{133,857} \approx 0.049 \] ### **Final Answer (b):** \[ \boxed{0.049} \] --- ### (c) Probability that an individual died from cancer and was a current cigar smoker \[ P(\text{died from cancer and current cigar smoker}) = \frac{\text{Number who died from cancer and were current cigar smokers}}{\text{Total number of individuals}} = \frac{163}{133,857} \approx 0.001 \] ### **Final Answer (c):** \[ \boxed{0.001} \] --- ### (d) Probability that an individual died from cancer or was a current cigar smoker #### **Step 1: Use the formula for the union of two events:** \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] where: - \( A \) = died from cancer - \( B \) = current cigar smoker #### **Step 2: Substitute the values:** \[ P(A) = 0.006, \quad P(B) = 0.049, \quad P(A \cap B) = 0.001 \] \[ P(A \cup B) = 0.006 + 0.049 - 0.001 = 0.054 \] ### **Final Answer (d):** \[ \boxed{0.054} \]