Calculate probabilities of compound events

Compound-event probability is determined by identifying the relevant outcomes and combining probabilities through addition for “or” events and multiplication for independent or conditional “and” events. Learners distinguish mutually exclusive events from overlapping events, avoid double-counting shared outcomes, and represent relationships with sample-space tables, organized lists, Venn diagrams, or tree diagrams; this scope does not extend to formal axiomatic probability, continuous distributions, or advanced counting theory.

Detailed Explanation: 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:

The probability of rolling an even number and getting heads.
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.

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