Skill: Calculate probabilities of compound events

A compound event combines two or more events.

• For “and,” both events must happen. For independent events, multiply:

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

• For “or,” at least one event must happen. If the events overlap, subtract the overlap so it is not counted twice:

$$P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$$

Example

A fair six-sided die is rolled and a fair coin is flipped.

Find:

1. The probability of rolling an even number and getting heads.
2. The probability of rolling an even number or getting heads.

Let:

• $E$ = rolling an even number
• $H$ = getting heads

The possible die results are $1,2,3,4,5,6$, so

$$P(E)=\frac{3}{6}=\frac12$$

because the even numbers are $2,4,6$.

For a fair coin,

$$P(H)=\frac12$$

1. Even and heads

The die roll and coin flip are independent, so multiply their probabilities:

$$P(E\text{ and }H)=P(E)\times P(H)$$ $$P(E\text{ and }H)=\frac12\times\frac12=\frac14$$

So, the probability is

$$\boxed{\frac14}$$

2. Even or heads

The events overlap: it is possible to roll an even number and get heads. Therefore, subtract the overlap:

$$P(E\text{ or }H)=P(E)+P(H)-P(E\text{ and }H)$$ $$P(E\text{ or }H)=\frac12+\frac12-\frac14$$ $$P(E\text{ or }H)=\frac34$$

So, the probability is

$$\boxed{\frac34}$$

The subtraction is important because the outcomes with an even roll and heads would otherwise be counted twice.