Two boxes are labelled I and II. Box I contains 4 black balls and 6 white balls.

Question

Two boxes are labelled I and II. Box I contains 4 black balls and 6 white balls. Box II contains 2 black balls and 8 white balls. A player randomly draws one ball from each box (so that a total of 2 balls are obtained).

1) What is the probability of obtaining at least one black ball?

2) What is the probability of obtaining exactly one black ball?

3) What is the probability of obtaining no black ball?

4) What is the probability of obtaining 2 black balls?

5) Given that at least one black ball is obtained, what is the probability that exactly one black ball is obtained?

6) Given that at least one black ball is obtained, what is the probability that the ball from box I is black?

7) Given that exactly one black ball is obtained, what is the probability that the ball from box I is black?

8) Let X be the number of black balls obtained by the player. Find the probability mass function of X.

9) Find the mean and variance of X.

Explanation / Answer

1.

P(at least one black) = 1 - P(both white)

= 1 - (6/10)*(8/10)

= 0.52 [ANSWER]
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2.

P(eactly one black) = P(black then white) + P(white then black)

= (4/10)*(8/10) + (6/10)*(2/10)

= 0.44 [ANSWER]

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3.

P(no black) = P(both white) = (6/10)*(8/10)

= 0.48 [ANSWER]

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4.

P(both black) = (4/10)*(2/10) = 0.08 [ANSWER]

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