Problem 82P

A and B flip coins. A starts and continues flipping until a tail occurs, at which point B starts flipping and continues until there is a tail. Then A takes over, and so on. Let P1 be the probability of the coin landing on heads when A flips and P2 when B flips. The winner of the game is the first one to get

(a) 2 heads in a row;

(b) a total of 2 heads;

(c) 3 heads in a row;

(d) a total of 3 heads.

In each case, find the probability that A wins.

Solution

Step 1 of 4

Given that A and B flip the coins

A start flipping the coin until tail occur then B takes over

B start flipping the coin until tail occur then A takes over

When A flips the coin is the probability of getting head

When B flips the coin is the probability of getting head

a) We have to find the probability of A wins when 2 heads in a row

Here we want the probability that A gets 2 heads before B gets

To find this

let A be the event that A gets 2 heads in row before B where A goes first means A wins

let B be the event that B gets 2 heads in row before A where B goes first means B wins

To find P(A) by conditioning on the output of the first 2 flips

To find that let us introduce the events that what is chance of happening in the first 2 flips

Let are the events of A’s coin lands heads up or tails up

And in the same way

Let are the events of B’s coin lands heads up or tails up

Now the conditioning on the first 2 flips

=

=

=…………(1)

Here because if a flips the coin goes to B

In the above expression we need

Now the conditioning on the first 2 flips

=

=

=

Substitute this expression in the equation (1) and get the value of

Now

=