Lesson 33: Finding the Probability of Independent and Dependent Events

New Concept

Events can be independent (unaffected by previous outcomes) or dependent (affected by previous outcomes), each with a distinct method for calculating combined probability.

What’s next

Next, you'll use this foundation to solve problems with tree diagrams, direct calculation, and the concept of odds.

Property

The outcome of the first event does not affect the second event. The probability is calculated as $P(A \text{ and } B) = P(A) \cdot P(B)$.

Examples

• Rolling a 5 on a die and flipping heads: $P(5 \text{ and heads}) = \frac{1}{6} \cdot \frac{1}{2} = \frac{1}{12}$.
• Drawing a card, replacing it, then drawing another: $P(\text{King and then Queen}) = \frac{4}{52} \cdot \frac{4}{52} = \frac{1}{169}$.
• A spinner with 8 sectors is spun twice: $P(\text{red and then blue}) = \frac{1}{8} \cdot \frac{1}{8} = \frac{1}{64}$.

Explanation

Think of this as totally separate actions, like flipping a coin and then rolling a die. What happens first has zero impact on what happens next! To find the combined chance of both happening, you simply multiply their individual probabilities together. It’s like a probability team-up where each member acts alone.

Property

The outcome of the first event does affect the second event. The probability is $P(A \text{ and } B) = P(A) \cdot P(B)$, where $P(B)$ is calculated under the new conditions.

Examples

• From a bag with 5 red and 4 blue marbles, drawing two red without replacement: $P(\text{red, then red}) = \frac{5}{9} \cdot \frac{4}{8} = \frac{20}{72} = \frac{5}{18}$.
• Drawing two face cards from a deck without replacement: $P(\text{Face Card, then Face Card}) = \frac{12}{52} \cdot \frac{11}{51} = \frac{132}{2652} = \frac{11}{221}$.
• Choosing two students from 10 girls and 8 boys: $P(\text{girl, then boy}) = \frac{10}{18} \cdot \frac{8}{17} = \frac{80}{306} = \frac{40}{153}$.

Explanation

Imagine grabbing a doughnut from a box and eating it. Now there's one less! The next person's choice is different because of what you did. The events are linked, so the probability changes for the second event based on the first outcome. It's a chain reaction of chances!