Story of Conditionnal Probability and Baye's Theorm

Conditional Probability is the probability of an event occurring given that another event has already occurred. To explain this concept using a story, let's consider a scenario where you are a weatherman. You want to predict whether it will rain tomorrow, but you know that the probability of it raining depends on whether there is a thunderstorm today or not. So, you calculate the Conditional Probability of rain tomorrow given that there was a thunderstorm today.

Bayes' Rule is used to calculate Conditional Probability. Bayes' Rule states that the Conditional Probability of event A given event B is equal to the probability of event B given event A, multiplied by the probability of A and divided by the probability of B. This may sound confusing, but let's see how this rule applies to our weatherman scenario.

Let's say that on a given day, there is a 20% chance of a thunderstorm occurring. If there is no thunderstorm, there is a 10% chance of rain. However, if there is a thunderstorm today, there is a 60% chance of rain tomorrow. We want to know the probability of rain tomorrow, given that there is a thunderstorm today.

Using Bayes' Rule, we can calculate the Conditional Probability of rain tomorrow given that there is a thunderstorm today:

P(Rain tomorrow | Thunderstorm today) = P(Thunderstorm today | Rain tomorrow) * P(Rain tomorrow) / P(Thunderstorm today)

We know that P(Rain tomorrow) = 0.2 0.6 + 0.8 0.1 = 0.26, and P(Thunderstorm today) = 0.2. But what is P(Thunderstorm today | Rain tomorrow)? We know that if there is a thunderstorm today, there is a 60% chance of rain tomorrow, so we can calculate P(Thunderstorm today | Rain tomorrow) as:

P(Thunderstorm today | Rain tomorrow) = P(Rain tomorrow | Thunderstorm today) * P(Thunderstorm today) / P(Rain tomorrow)
= 0.6 * 0.2 / 0.26 = 0.4615

Now, we can substitute all our values into Bayes' Rule and calculate the Conditional Probability of rain tomorrow given that there is a thunderstorm today:

P(Rain tomorrow | Thunderstorm today) = 0.6 * 0.2 / 0.26
= 0.4615

Therefore, there is a 46.15% chance of rain tomorrow, given that there is a thunderstorm today.