Find the probability of three balls

Question

A bag contains 4 White and 7 Black balls. Three balls are selected without replacement.

A = 1 W 2B

B = at least one black ball

 

a. Find P(A)

b. Find P(B)

c. Are A and B independent?

d. Find P(A ∪ B)

Answer 1

Hi Jean,

 

a) Probabilities under sampling without replacement follow the hypergeometric distribution: number of ways to select 3 balls is 11C3, and the number of ways to select 1W 2B is 4C1 × 7C2. The probability is 4C1 × 7C2 / 11C3.

 

b) The easiest way to do this is to take 1 - P (0 black balls).

 

c) Compare P (B | A) with your answer to b). If they are the same, then A and B are independent, otherwise A and B are dependent. As a hint, you don't need to do a calculation to find P (B | A), simply think about the meaning of the event B | A.

 

d) Use P (A ∪ B) = P (A) + P (B) - P (A ∩ B). You have the first two parts from a) and b). You don't need to do any calculation to find P (A ∩ B), simply think about the meaning of the event A ∩ B.