13  Introduction

13.1 Probability of a Conjunction of Two Events

The probability of the conjunction (intersection) of two independent events \(A\) and \(B\) can be calculated using the formula:

\[ \text{Probability} (A \text{ and } B) = \text{Probability} (A) \times \text{Probability} (B) \]

This formula assumes that events $ A $ and $ B $ are independent, meaning the occurrence of one event does not affect the occurrence of the other.

For example, if you want to find the probability of rolling a 3 on a fair six-sided die (Event $ A $) and flipping heads on a fair coin (Event $ B $), and both events are independent, you would first find the individual probabilities of each event:

• Probability of rolling a 3 on a six-sided die: $ P(A) = $ (1 favorable outcome out of 6 possible outcomes)
• Probability of flipping heads on a fair coin: $ P(B) = $ (1 favorable outcome out of 2 possible outcomes)

Then, you can calculate the probability of the conjunction of these events:

\[ \text{Probability} (A \text{ and } B) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \]

So, the probability of rolling a 3 on a fair six-sided die and flipping heads on a fair coin is $ $ or approximately 0.0833 when both events are independent.