Let's break this down step by step: --- # 1. Probability Tree for the Experiment ### Description - On each cast, the fisherman can either **catch a fish** (with probability \( p \)) and stop, or **not catch a fish** (with probability \( 1-p \)) and continue to the next cast. - He repeats this until he catches a fish. ### Tree Structure (First Few Steps) - **Start** - **Catch on 1st Cast**: Probability = \( p \) → **Stop** - **No Catch on 1st Cast**: Probability = \( 1-p \) → Try again - **Catch on 2nd Cast**: Probability = \( p \) → **Stop** - **No Catch on 2nd Cast**: Probability = \( 1-p \) → Try again - **Catch on 3rd Cast**: Probability = \( p \) → **Stop** - ... and so on Here's a visualization (truncated to the first three casts): ``` Start | |-- Catch on 1st try (prob = p) --> End | |-- No Catch (prob = 1-p) | |-- Catch on 2nd try (prob = p) --> End | |-- No Catch (prob = 1-p) | |-- Catch on 3rd try (prob = p) --> End |-- ... (continues similarly) ``` --- # 2. Probabilities for Each Case Let \( C_i \) be the event that the fisherman catches a fish for the first time on the **i-th** cast. ## Find \( P(C_1) \), \( P(C_2) \), and the General Expression \( P(C_n) \) ### Probability on the First Cast (\( C_1 \)) - **Catches fish immediately:** \[ P(C_1) = p \] --- ### Probability on the Second Cast (\( C_2 \)) - **Misses first cast** (\( 1-p \)), then **catches on second** (\( p \)): \[ P(C_2) = (1-p) \cdot p \] --- ### Probability on the n-th Cast (\( C_n \)) - **Misses first \( n-1 \) casts** (\( (1-p)^{n-1} \)), then **catches on the n-th cast** (\( p \)): \[ P(C_n) = (1-p)^{n-1} \cdot p \] --- ## **Summary Table** | Cast (i) | Probability \( P(C_i) \) | |-----------|--------------------------------| | 1 | \( p \) | | 2 | \( (1-p)p \) | | n | \( (1-p)^{n-1} p \) | --- ## **Key Point** This is the **geometric distribution**: - The probability of the first success on the \( n \)-th trial. --- If you need more details or want to see the tree drawn more visually, let me know!