Lesson 88: Independent and Dependent Events
When calculating probabilities of combined events, it is crucial to determine whether the events are independent or dependent. This distinction changes how probabilities are computed.
Definitions
- Independent Events: Two events are independent if the outcome of one does not affect the outcome of the other. Example: Flipping a coin and rolling a die.
- Dependent Events: Two events are dependent if the outcome of one affects the probability of the other. Example: Drawing two cards from a deck without replacement.
Probability Rules
- Independent: \( P(A \cap B) = P(A) \times P(B) \)
- Dependent: \( P(A \cap B) = P(A) \times P(B|A) \) (where \( P(B|A) \) is the probability of B given that A has occurred).
Examples
Example 1 — Independent:
Flip a coin and roll a die. What is the probability of getting heads and a 6? \( P(H) = 1/2 \), \( P(6) = 1/6 \). \( P(H \cap 6) = (1/2)(1/6) = 1/12 \).
Example 2 — Dependent:
Draw two cards without replacement. What is the probability both are queens? \( P(\text{Queen first}) = 4/52 \). \( P(\text{Queen second | Queen first}) = 3/51 \). \( P = (4/52)(3/51) = 12/2652 = 1/221 \).
Example 3 — Dependent:
A jar contains 5 red and 3 blue marbles. Two are drawn without replacement. Probability both are red: \( P = (5/8)(4/7) = 20/56 = 5/14 \).
Practice Problems
- Flip a coin and roll a die. Find the probability of tails and an odd number.
- Draw two cards with replacement. Find the probability both are hearts.
- Draw two cards without replacement. Find the probability both are aces.
- A bag has 6 red and 4 green marbles. Two marbles are drawn without replacement. Find the probability both are green.
- Roll two dice. Find the probability both show a 5.