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A toss the coin first, B toss the coin second, and C tosses third. With the number of outcomes repeated, we have proof of the probability that each of the three players is winning.

Every subset of a sample space of a random experiment is called an event.

The events are generally denoted by capital letters such as A, B, C, etc.

Consider A player wins the probability that he won on his first toss is\(\frac{1}{2}\).

The second toss of probability won is

\(\begin{aligned}{} &= \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\\ &= {\left( {\frac{1}{2}} \right)^4}\end{aligned}\)

The first three\(\frac{{\bf{1}}}{{\bf{2}}}\)鈥檚 are the probability that A got tails, then B got tails, and then C got tails so that A will be tossed repeated.

Using the same procedure, we see that the probability of player A winning on his third flip is \({\left( {\frac{1}{2}} \right)^7}\).

We see that the player A鈥檚 total probability of wining is

\(\frac{1}{6}\)\(\frac{1}{2},{\left( {\frac{1}{2}} \right)^4},{\left( {\frac{1}{2}} \right)^7},.....{\left( {\frac{1}{2}} \right)^{3n - 2}}\)

The geometric progression with \(a = \frac{1}{2}\) and \(r = \frac{1}{8}\). That means that the sum is \(\begin{aligned}{}S = \frac{a}{{1 - r}} = \frac{{\frac{1}{2}}}{{1 - \frac{1}{8}}}\\ = \frac{4}{7}\end{aligned}\)

To represent the P(A), the probability of player B winning:

\(\begin{aligned}{}p\left( C \right) &= \frac{1}{2} \times \frac{2}{7}\\ &= \frac{1}{7}\end{aligned}\)

\(\begin{aligned}{}S &= \frac{a}{{1 - r}} &= \frac{2}{{1 - \frac{1}{8}}}\\ &= \frac{2}{7}\end{aligned}\)

To represents the P(B), the probability of player A winning.