Question

9.74. Suppose we toss a biased coin independently until we get two heads or two tails in total. The coin produces a head with probability p on any toss.

1. What is the sample space of this experiment?
2. What is the probability function?
3. What is the probability that the experiment stops with two heads?

Answer 1

9.74)

Now, P(head) = p and P(tail) = 1-p

1)

The Sample space is given by:

$S=\begin{Bmatrix} HH,TT,HTH,THH,THT,HTT \end{Bmatrix}$

2)

$P(HH)=p.p=p^2,P(TT)=(1-p)(1-p)=(1-p)^2$

$P(HTH)=p(1-p)p=p^2(1-p),P(THH)=(1-p)p.p=p^2(1-p)$

P(THT) = (1-P)p(1-P) = p(1 - p), P(HTT) = p(1 - p)(1 - p) = p(1-P)

So, the probability function is given by:

X	P(X)
HH	$p^2$
TT	$(1-p)^2$
HTH	$p^2(1-p)$
THH	$p^2(1-p)$
THT	$p(1-p)^2$
HTT	$p(1-p)^2$

3)

Required probability = P(experiment stops with two heads)

= P(HH) + P(HTH) + P(THH)

$p^2+p^2(1-p)+p^2(1-p)=p^2(3-2p)$