Lecture Notes: Probability of Drawing Queens from a Deck
Overview
This lecture addresses finding the probability of drawing two queens from a standard 52-card deck in two scenarios:
- Drawing with replacement.
- Drawing without replacement.
Definitions
- Q1: Event where the first card drawn is a queen.
- Q2: Event where the second card drawn is a queen.
Probability with Replacement
- Independence of Events: When drawing with replacement, the events Q1 and Q2 are independent.
- Calculation:
- Probability of Q1 is 4/52 (since there are 4 queens in a 52 card deck).
- Replace the queen and shuffle the deck.
- Probability of Q2 remains 4/52 (since the deck is unchanged).
- Combined probability: ( \frac{4}{52} \times \frac{4}{52} = 0.0059 )
- Result: Less than 1% chance of drawing two queens with replacement.
Probability without Replacement
- Dependence of Events: When drawing without replacement, the events Q1 and Q2 are dependent.
- Calculation:
- Probability of Q1 is still 4/52.
- After drawing a queen, do not replace it, leaving 3 queens in a 51-card deck.
- Probability of Q2 is now 3/51.
- Combined probability: ( \frac{4}{52} \times \frac{3}{51} = 0.0045 )
- Result: Less than 1% chance of drawing two queens without replacement, slightly less than with replacement.
Conclusion
- Probability differs slightly between drawing with replacement and without replacement.
- Both probabilities are less than 1%, indicating a low likelihood of drawing two queens consecutively under either scenario.