Compound Events and Probability

A compound event involves finding the probability of two or more simple events happening together. For example, flipping a coin and rolling a die, or drawing two cards from a deck.

To find the probability of compound events, you need to know all the possible outcomes.

Finding All Possible Outcomes

The list of all possible outcomes for an experiment is called the sample space. You can find the sample space using a few different methods:

• Organized Lists: Write down combinations systematically. For flipping two coins, the list is: (Heads, Heads), (Heads, Tails), (Tails, Heads), (Tails, Tails).
• Tree Diagrams: A visual tool where each branch represents a possible outcome. If you spin a spinner with 3 colors twice, you draw 3 branches for the first spin, and from the end of each of those, 3 more branches for the second spin, giving $3 \times 3 = 9$ total outcomes.

Independent Events

When the outcome of the first event does not affect the outcome of the second event, they are called independent events.

To find the probability of two independent events happening together, you simply multiply their individual probabilities:

$P(A \text{ and } B) = P(A) \times P(B)$

Example: Flipping Two Heads

What is the probability of flipping two heads in a row?

1. The probability of getting heads on the first flip: $P(\text{Heads}) = \frac{1}{2}$
2. The probability of getting heads on the second flip: $P(\text{Heads}) = \frac{1}{2}$
3. Multiply them: $P(\text{Two Heads}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Using the Sample Space

Sometimes, it is easier to look at the total sample space to find a probability, especially when adding numbers together.

Example: Rolling a Sum of 7

What is the probability that two standard six-sided dice sum to 7?

1. Find total outcomes: Each die has 6 sides, so there are $6 \times 6 = 36$ possible combinations.
2. Find favorable outcomes: Which combinations add up to 7?
   • (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)
   • There are exactly 6 favorable outcomes.
3. Calculate probability: Divide the favorable outcomes by the total outcomes.

$P(\text{Sum of 7}) = \frac{6}{36} = \frac{1}{6}$

By organizing your outcomes and knowing when to multiply, you can easily calculate the probability of any compound event!