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There is a 30% chance of rain and a 70% ...

There is a `30%` chance of rain and a `70%` chance of shine. If it rains, there is a `50%` chance that Bob will cancel his picnic, but if the sun is shining, he will definitely have his picnic. What is the chance that Bob will have his picnic?

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To find the probability that Bob will have his picnic, we can break down the problem step by step. ### Step-by-Step Solution: 1. **Identify the probabilities given in the question:** - Probability of rain (P(Rain)) = 30% = 0.3 - Probability of sunshine (P(Sunshine)) = 70% = 0.7 - Probability that Bob cancels his picnic if it rains (P(Cancel | Rain)) = 50% = 0.5 - Probability that Bob has his picnic if it is sunny (P(Picnic | Sunshine)) = 100% = 1 2. **Calculate the probability that Bob will cancel his picnic if it rains:** - If it rains, the probability that Bob will **not** have his picnic is: \[ P(No Picnic | Rain) = P(Cancel | Rain) = 0.5 \] - Therefore, the probability that Bob will have his picnic if it rains is: \[ P(Picnic | Rain) = 1 - P(No Picnic | Rain) = 1 - 0.5 = 0.5 \] 3. **Calculate the total probability that Bob will have his picnic:** - We can use the law of total probability: \[ P(Picnic) = P(Picnic | Rain) \cdot P(Rain) + P(Picnic | Sunshine) \cdot P(Sunshine) \] - Substitute the known values: \[ P(Picnic) = (0.5 \cdot 0.3) + (1 \cdot 0.7) \] - Calculate each term: \[ P(Picnic) = 0.15 + 0.7 = 0.85 \] 4. **Convert the probability to percentage:** - To express the probability as a percentage: \[ P(Picnic) = 0.85 \times 100\% = 85\% \] ### Final Answer: The probability that Bob will have his picnic is **85%**.
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