Mathematics

Problem

A game produces four mutually exclusive outcomes whose probabilities depend on a parameter

k

as shown below: 1.

A\cap B

occurs with probability

k

, 2.

A\cap B^{c}

occurs with probability

0.3

, 3.

A^{c}\cap B

occurs with probability

0.2

, 4.

A^{c}\cap B^{c}

occurs with probability

0.5-k

. Here

A

denotes that the player scores at least

2

points, while

B

denotes that the player uses a bonus card during the turn. a. (a) Using the fact that the total probability of all outcomes is

1

, state the range of permissible values of

k

. b. (b) Compute

P(A)

and

P(B)

in terms of

k

and hence derive an equation that must be satisfied if

A

and

B

are to be independent. c. (c) Solve the equation obtained in part (b) to find all permissible values of

k

that make

A

and

B

independent. d. (d) For each value of

k

found in part (c), calculate

P(A\cap B)

and comment on whether the independence condition is met in practice.