Compound events happen when we look at two or more things at the same time. It's important to know how these events affect probability, especially when we’re in a gym or similar setting.

Let’s take a look at two events:

• Event A: Rolling a die and getting an even number.
• Event B: Flipping a coin and getting heads.

For Event A, the possible even numbers we can roll on a die are {2, 4, 6}. For Event B, the outcomes when flipping a coin are {Heads, Tails}. To find the chance of both events happening (A and B), we need to look at each event separately and then bring them together.

Calculating Probability

1. Chance of Event A (even number): There are 3 good outcomes (2, 4, 6) out of 6 total options when we roll a die. $P(A) = \frac{3}{6} = \frac{1}{2}$

2. Chance of Event B (heads): There is 1 good outcome (Heads) out of 2 total options when we flip a coin. $P(B) = \frac{1}{2}$

Now we’ll find the chance of both events happening at the same time (A and B). If we assume these events don’t affect each other, we can multiply their probabilities: $P(A \, \text{and} \, B) = P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$

3. Combined Outcomes: To understand this better, we can list all possible outcomes that include both events:

• (2, Heads)
• (2, Tails)
• (4, Heads)
• (4, Tails)
• (6, Heads)
• (6, Tails)

With this example, we see how compound events add extra steps in probability calculations. Learning how to find probabilities in these kinds of situations is key to really getting the hang of probability in everyday life!