Probability: Two Events
Understand independent and dependent events, calculate combined probabilities, and use tree diagrams to organise outcomes.
Independent and Dependent Events
When two events occur together, we need to know whether the outcome of one event affects the other. This determines which probability rule to use.
Independent Events
The outcome of one event does not affect the probability of the other. Replacement restores independence.
Examples:
- Flipping a coin twice
- Rolling a die, then flipping a coin
- Drawing a card with replacement
Dependent Events
The outcome of one event does affect the probability of the other. No replacement creates dependence.
Examples:
- Drawing two cards without replacement
- Selecting two students from a class
- Choosing marbles without replacing them
The Multiplication Rule
To find the probability that both events A and B occur, multiply their individual probabilities. The formula depends on whether the events are independent or dependent.
Independent Events
P(A and B) = P(A) × P(B)
Dependent Events
P(A and B) = P(A) × P(B|A)
P(B|A) = probability of B given A has occurred
Remember: All probabilities must be between 0 and 1. All probabilities in a complete sample space must add to 1.
Tree Diagrams
A tree diagram is a visual tool that lists all possible outcomes of two or more events. Each branch shows one possible result, and probabilities along branches are multiplied to find the combined outcome probability.
Tree diagram for flipping a coin twice. Each outcome has probability 1/4.
Key rule: Multiply probabilities along branches to find the probability of that path. Add probabilities of separate paths to find P(combined outcome).
Key Vocabulary
| Term | Definition |
|---|---|
| Independent events | Two events where the outcome of one does not affect the probability of the other. |
| Dependent events | Two events where the outcome of the first affects the probability of the second. |
| Conditional probability | P(B|A) is the probability of B occurring given that A has already occurred. |
| Tree diagram | A branching diagram used to list all possible outcomes of two or more events. |
Worked Examples
Independent Events
A fair coin is flipped and a die is rolled. Find P(Head and 6).
These events are independent (one does not affect the other).
P(Head) = 1/2 P(6) = 1/6
P(Head and 6) = 1/2 × 1/6 = 1/12
Dependent Events (Without Replacement)
A bag has 5 red and 3 blue marbles. Two marbles are drawn without replacement. Find P(both red).
P(1st red) = 5/8
P(2nd red | 1st red) = 4/7 (only 4 red remain from 7)
P(both red) = 5/8 × 4/7 = 20/56 = 5/14
Using a Tree Diagram
A student guesses on two true/false questions. What is P(at least one correct)?
Outcomes: CC (1/4), CW (1/4), WC (1/4), WW (1/4)
P(at least one correct) = P(CC) + P(CW) + P(WC) = 1/4 + 1/4 + 1/4 = 3/4
Or: P(at least one) = 1 − P(none correct) = 1 − 1/4 = 3/4
Knowledge Check
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Key Concepts Summary
- ●Independent events: P(A and B) = P(A) × P(B). The outcome of one does not affect the other.
- ●Dependent events: P(A and B) = P(A) × P(B|A). Sampling without replacement creates dependence.
- ●Tree diagrams: multiply along branches for each path; add paths together for combined outcomes.
- ●All probabilities must be between 0 and 1, and all outcomes in the sample space must sum to 1.