Question

Suppose that the probability of a head on any toss of a certain coin isp(0<p <1), and suppose that the coin is tossed repeatedly. LetX_ndenote the total number ofheads that have been obtained on the firstntosses, and letY_n=n−X_ndenote the total number of tails on the firstntosses. Suppose that the tosses are stopped as soon as a numbernis reached such that eitherX_n=Y_n+3 orY_n=X_n+3. Determine the probability thatX_n=Y_n+3when the tosses are stopped.

Short answer

If tosses are stopped the probability that\({X_n} = {Y_n} + 3\)is\(\frac{1}{2}\)

Step-by-step solution

Given information

Probability of a certain coin being head in a toss is\(p;\left( {0 < p < 1} \right)\).

Total no. of head obtained on the first n tosses is \({X_n}\).

State the events

Let, A win one dollar if head obtained in a toss otherwise B win one dollar if tail occurs.

Assuming that the initial fortune of two persons A and B are 3 dollars.

Then, if\({X_n} = {Y_n} + 3\)then A won the game and if\({Y_n} = {X_n} + 3\)then B won the game.

So, here \(i = 3\)and \(k = 6\)

Compute the probability

Referring to equation 2.4.6 for the following equation.

If\(p = \frac{1}{2}\), then the probability is given by,

\(\begin{aligned}{c}a = \frac{i}{k}\\ = \frac{3}{6}\\ = \frac{1}{2}\end{aligned}\)

If\(p \ne \frac{1}{2}\)then the probability is given by,

\(\frac{1}{{{{\left( {\frac{{1 - p}}{p}} \right)}^3} + 1}}\)

Hence, if tosses are stopped the probability that \({X_n} = {Y_n} + 3\) is \(\frac{1}{2}\).