Question

Three players A, B, and C, take turns tossing a fair coin. Suppose that A tosses the coin first, B tosses second, and C tosses third; suppose that this cycle is repeated indefinitely until someone wins by being the first player to obtain ahead. Determine the probability that each of the three players will win.

Short answer

In the end, the probability is 1

Step-by-step solution

Given Information

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.

State the given events and compute the probability

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\(\frac{{\bf{1}}}{{\bf{2}}}\)’s 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 \({\left( {\frac{1}{2}} \right)^7}\).

Step 3) We have proof by geometric progression.

We see that the player A’s 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.