Mathematics

Problem

A biased coin shows Head with probability

p

on every toss, where

0<p<1

. The coin is tossed three times. Define the events A : “exactly two Heads occur”, B : “the first toss is Tail”. Answer the following questions. (i) Express

P(A)

and

P(B)

in terms of

p

. (ii) Obtain

P(A\cap B)

in terms of

p

. (iii) Find all value(s) of

p

for which events

A

and

B

are independent. (iv) Investigate whether events

A

and

B

can ever be mutually exclusive. Provide full reasoning.

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