Probability ③ Conditional Probability and Multiplication Theorem
P(A|B) = P(A∩B)/P(B) (the probability of A given that B has occurred)
Multiplication theorem: P(A∩B) = P(B)×P(A|B)
Example ①
A bag contains 4 red balls and 3 white balls. Two balls are drawn without replacement. Probability that both are red:
P = 4/7 × 3/6 = 2/7
A bag contains 4 red balls and 3 white balls. Two balls are drawn without replacement. Probability that both are red:
P = 4/7 × 3/6 = 2/7
Example ②
One number was selected from 1 to 10 and was even. Probability that it is a multiple of 4:
P(multiple of 4|even)=(2/10)/(5/10) = 2/5
One number was selected from 1 to 10 and was even. Probability that it is a multiple of 4:
P(multiple of 4|even)=(2/10)/(5/10) = 2/5
Key Points
- Sampling without replacement → multiplication theorem
- P(A|B) ≠ P(B|A) (not symmetric)
Practice Problems
- Given that at least one head appears when two coins are tossed, what is the probability that both are heads?
- Two balls are drawn without replacement from a bag containing 5 red balls and 3 white balls. What is the probability that both are white?
- When P(A)=1/3, P(B)=1/2, and P(A∩B)=1/6, find P(A|B) and P(B|A).
Answers
- (1/4)/(3/4)=1/3
- 3/8×2/7=6/56=3/28
- P(A|B)=(1/6)/(1/2)=1/3、P(B|A)=(1/6)/(1/3)=1/2