Answer:
The probability of A winning is [tex]\frac{1}{2-p}[/tex] and the probability of B winning is [tex]\frac{1-p}{2-p}[/tex].
Step-by-step explanation:
The probability of winning is p.
The game stops when either A or B wins.
The sample space of A winning is as follows:
{S, FFS, FFFFS, FFFFFFS, ...}
The sample space of B winning is as follows:
{FS, FFFS, FFFFFS, FFFFFFFS, ...}
Compute the probability of A winning as follows:
P (A winning) = P (S) + P (FFS) + P (FFFFS) + ...
[tex]=p+(1-p)(1-p)p+(1-p)(1-p)(1-p)(1-p)p+...\\=p\sum\limits^{\infty}_{i=0} {(1-p)^{2i}}\\=p\times \frac{1}{1-(1-p)^{2}}\\=\frac{p}{1-(1-p)^{2}}\\=\frac{p}{1-1-p^{2}+2p}\\=\frac{1}{2-p}[/tex]
Compute the probability of B winning as follows:
P (B winning) = P (FS) + P (FFFS) + P (FFFFFS) + ...
[tex]=(1-p)p+(1-p)(1-p)(1-p)p+(1-p)(1-p)(1-p)(1-p)(1-p)p+...\\=(1-p)p\sum\limits^{\infty}_{i=0} {(1-p)^{2i}}\\=(1-p)p\times \frac{1}{1-(1-p)^{2}}\\=\frac{(1-p)p}{1-(1-p)^{2}}\\=\frac{(1-p)p}{1-1-p^{2}+2p}\\=\frac{1-p}{2-p}[/tex]
Thus, the probability of A winning is [tex]\frac{1}{2-p}[/tex] and the probability of B winning is [tex]\frac{1-p}{2-p}[/tex].