Section 15.5: Independent and Dependent Events

Property

A compound event is where a particular experiment is executed two or more times. An important factor is the distinction between independent and dependent outcomes. Two events are independent if the occurrence or nonoccurrence of one event has no effect on the other. In contrast, a dependent event is a second event whose result depends on the result of a first event.

Examples

• Rolling a die and then spinning a spinner are independent events. The die roll does not change the probabilities on the spinner.
• A box has 5 red and 4 blue pens. Drawing a red pen is the first event. If you don't replace it, the probability of drawing a second red pen changes from $\frac{5}{9}$ to $\frac{4}{8}$, so the events are dependent.
• Flipping a coin twice are independent events. If you get heads the first time, the probability of getting heads the second time is still $\frac{1}{2}$.

Explanation

Two events are independent if the first outcome doesn't affect the second (like two coin flips). They are dependent if the first outcome changes the possibilities for the second (like drawing a card and not putting it back).

Property

Find probabilities of compound events by identifying whether the events are independent or dependent. For independent events (where one event does not affect the other), multiply the individual probabilities: $P(A \text{ and } B) = P(A) \times P(B)$. For dependent events (where the first event affects the second), use conditional probability: $P(A \text{ and } B) = P(A) \times P(B|A)$, where $P(B|A)$ is the probability of B given that A has occurred.

Examples

• Independent: The probability of rolling a 4 on a die ($\frac{1}{6}$) and flipping heads on a coin ($\frac{1}{2}$) is $\frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$.
• Independent: A bag has 3 red and 2 blue marbles. The probability of drawing a red marble, replacing it, and then drawing a blue marble is $\frac{3}{5} \times \frac{2}{5} = \frac{6}{25}$.
• Dependent: A bag has 3 red and 2 blue marbles. The probability of drawing a red marble, not replacing it, and then drawing a blue marble is $\frac{3}{5} \times \frac{2}{4} = \frac{6}{20} = \frac{3}{10}$.

Explanation

The key to calculating compound event probabilities is determining whether the events are independent or dependent. Independent events don't influence each other, so you simply multiply their individual probabilities. Dependent events affect each other, so the probability of the second event changes based on the outcome of the first event.

Property

Conditional probability is the probability of event B occurring given that event A has already occurred, written as $P(B|A)$ and calculated as: