Question

q1)Suppose that we have two bags each containing black and white balls. One bag contains two times as many white balls as black balls. The other bag contains two times as many black balls as white. Suppose we choose one of these bags at random. For this bag we select five balls at random, replacing each ball after it has been selected. The result is that we find all balls are white balls. What is the probability that we were using the bag with mainly black balls?

q2)Given that event A happens, event B must happen. Which of the following statements is CORRECT?

a)P(A) > P(B)

b)There is not enough information to compare P(A) and P(B).

c)P(A) ≤ P(B)

d)P(A) ≥ P(B)

e)P(A) < P(B)

q3)

which of the following statements is/are correct?

1. If two events are independent, then the covariance must be zero

2. If the covariance of two events are zero, then the two events may be independent

3. If the correlation of two events are zero, then the two events may be dependent.

4. If the correlation of two events are zero, then the covariance may not be zero

Answer 1

1)P(all white) =P(mainly black)*P(5 white |mainly black)+P(mainly white)*P(5 white |mainly white)

=(1/2)*(1/3)^5+(1/2)*(2/3)^5 =11/162

P(black balls |all 5 white) =(1/2)*(1/3)^5/(11/162)=1/33

2)

d)P(A) ≥ P(B) (since event A contains B)

3)correct options are"

If two events are independent, then the covariance must be zero

1. If the covariance of two events are zero, then the two events may be independent

2. If the correlation of two events are zero, then the two events may be dependent.