Calculating Probability of Rain Using Bayes Theorem

Calculate the probability of it raining during a picnic given that it is cloudy in the morning using Bayes' Theorem. You know the probability of rain on any given day is 0.15, the probability of it being cloudy in the morning on any given day is 0.25, and the probability of it being cloudy if it rains is 0.8.

Bayes' Theorem is a powerful tool for calculating conditional probabilities, which are probabilities of an event occurring given that another event has already occurred. In this problem's context, we use it to find the probability of rain during a picnic given that it is cloudy in the morning. The idea is to update our belief about rain based on the new evidence about the weather condition in the morning. To solve this, we need to understand the relationship between the marginal probability of the initial events and their conditional probabilities. The problem provides the marginal probabilities such as the likelihood of rain on any day, and cloudiness in the morning, as well as the conditional probability of it being cloudy if it rains. Bayes' Theorem will allow us to rearrange these probabilities to find the desired conditional probability. The problem requires careful substitution into Bayes' Theorem formula, where we take into account the prior probability (rain on any day) and the likelihood (cloudy given rain) to compute the posterior probability (rain given cloudy morning). Such problems underline the significance of conditional probability in real-world situations where events are interdependent and involve updating our statistical understanding with new information.