How do you find the conditional probability from a given set of data?

To find conditional probability, divide the probability of both events happening by the probability of the given event.

Conditional probability helps us understand the likelihood of an event occurring given that another event has already happened. Let's say we have two events, A and B. The conditional probability of A given B is written as P(A|B). This is calculated using the formula:

\[ P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \]

Here's how you can do it step-by-step:

1. **Identify the Events**: Determine which events you are interested in. For example, A could be "it rains" and B could be "it is cloudy".

2. **Find the Joint Probability**: Calculate or find the probability that both events A and B happen together, written as P(A and B).

3. **Find the Probability of the Given Event**: Determine the probability of the event that is given, in this case, P(B).

4. **Apply the Formula**: Use the formula \( P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \) to find the conditional probability.

For example, if you have data showing that it rains on 30 out of 100 days, and it is cloudy on 50 out of those 100 days, and it rains and is cloudy on 20 out of those 100 days, you can find the conditional probability that it rains given that it is cloudy. Here, P(A and B) is 20/100, and P(B) is 50/100. So,

\[ P(\text{Rain}|\text{Cloudy}) = \frac{20/100}{50/100} = \frac{20}{50} = 0.4 \]

This means there is a 40% chance it will rain given that it is cloudy.