Skill: Calculate probabilities of independent events

To find the probability that two independent events both happen, multiply their probabilities:

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

Events are independent when one event does not change the probability of the other. For example, rolling a die does not affect the result of flipping a coin.

Example: A fair die is rolled and a fair coin is flipped. What is the probability of rolling an even number and getting heads?

1. Find the probability of each event.

   • Even numbers on a die are $2,4,6$, so

$$P(\text{even})=\frac{3}{6}=\frac{1}{2}$$

• A fair coin has one heads outcome out of two, so

$$P(\text{heads})=\frac{1}{2}$$

2. Check that the events are independent. The die roll does not affect the coin flip.

3. Multiply the probabilities:

$$P(\text{even and heads}) =\frac{1}{2}\times\frac{1}{2} =\frac{1}{4}$$

Therefore, the probability is $\boxed{\frac14}$, or $25\%$.

Remember: use multiplication for independent events joined by “and.” Independent events are not the same as mutually exclusive events. Mutually exclusive events cannot happen at the same time, while independent events do not affect each other.