in a certain city the weather on a day if reported as sunny ,cloudy or rainy,if a day is sunny,the probability that the next day is sunny is 30%,cloudy is 20%,and rainy is 50%. if the day is rainy the probability that the next day is sunny is 30%,cloudy is 30%,and rainy is 40%.if a sunday is sunny ,find the probability that the next tuesday is rainy.
To find the probability that the next Tuesday is rainy given that the current Sunday is sunny, we need to use conditional probability.
Let's break down the given information:
- There are three possible weather conditions: sunny, cloudy, and rainy.
- Given a sunny day, the probability of the next day being sunny is 30%, cloudy is 20%, and rainy is 50%.
- Given a rainy day, the probability of the next day being sunny is 30%, cloudy is 30%, and rainy is 40%.
We want to find the probability of the next Tuesday being rainy given that the current Sunday is sunny.
To solve this, we can use the law of total probability, which states that the probability of an event is equal to the sum of the probabilities of that event occurring given each different possible condition.
Let's calculate the probability step by step:
1. Determine the probability of having sunny weather on Monday given that Sunday is sunny:
- P(Monday sunny | Sunday sunny) = 0.3
2. Determine the probability of having cloudy weather on Monday given that Sunday is sunny:
- P(Monday cloudy | Sunday sunny) = 0.2
3. Determine the probability of having rainy weather on Monday given that Sunday is sunny:
- P(Monday rainy | Sunday sunny) = 0.5
4. Calculate the probability of having rainy weather on Tuesday given that Monday is sunny:
- P(Tuesday rainy | Monday sunny) = P(Tuesday rainy | Monday sunny and Sunday sunny) * P(Monday sunny | Sunday sunny)
- P(Tuesday rainy | Monday sunny) = 0.4 * 0.3
Therefore, the probability of the next Tuesday being rainy given that the current Sunday is sunny is 0.4 * 0.3 = 0.12, or 12%.