With explanation please Suppose that A tosses a coin which lands heads with prob

Question

With explanation please

Suppose that A tosses a coin which lands heads with probability p_A, and B tosses one which lands heads with probability p_B. They toss their coins simultaneously over and over again, in a competition to see who gets the first head. The one to get the first head is the winner, except that a draw results if they get their first heads together. Calculate the probability that A wins; the probability that B wins; the probability that a draw occurs; the distribution of the number of times A and B must toss.

Explanation / Answer

a) probabilty that A wins =P(in first attemt head comes+for A head not come then for B it does not come then it comes for A+......)

=pa+(1-pa)*(1-pb)pa+(1-pa)2*(1-pb)2pa+(1-pa)3*(1-pb)3pa+.......

   =pa/(1-(1-pa)(1-pb)=pa/(pa+pb-papb)

b) similarly probabilty for B wins=(1-pa)pb+(1-pa)2(1-pb)pb+(1-pa)3(1-pb)2pb+.......

=(1-pa)pb/(pa+pb-papb)

c) probabilty of draw =(1-pa)(1-pb)+(1-pa)2(1-pb)2+(1-pa)3(1-pb)3+(1-pa)4(1-pb)4+....

=(1-pa)(1-pb)/(pa+pb-papb)

d)distribution of number of tosses for A =f(X) =(1-pa)x-1*(1-pb)x-1pa

distribution of number of tosses for B =f(X) =(1-pa)x*(1-pb)x-1pb

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