Calculate probabilities of independent compound events

Independent events are outcomes in which the occurrence of one event does not change the probability of the other; for a compound event requiring both events, the probability is found by multiplying their probabilities, including across repeated trials. Tree diagrams, organized lists, and fractions or decimals can represent these products, while “either event” requires accounting for any overlap rather than simply adding blindly. Conditional probability and dependent events, such as sampling without replacement, are not included at this level.

Detailed Explanation: Calculate probabilities of independent compound events

When two events are independent, one event does not change the chance of the other. If a compound event requires both events to happen, multiply their probabilities:

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

Example

A fair six-sided die is rolled, and a fair coin is flipped. What is the probability of rolling an even number and flipping heads?

Step 1: Find the probability of each event.

The even numbers on the die are $2,4,$ and $6$ , so:

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

A fair coin has one heads outcome out of two:

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

Step 2: Check that the events are independent.

The die roll does not affect the coin flip, so the events are independent.

Step 3: Multiply the probabilities.

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

So, the probability of rolling an even number and flipping heads is:

\boxed{\frac{1}{4}}

For independent events that both must happen, remember: “and” means multiply.

Learn by doing: Calculate probabilities of independent compound events

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