First slide

Theorems of probability

Question

In a game a coin is tossed 2n + m times and a player wins if he does not get any two consecutive outcomes same for at least 2n times in a row. The probability that player wins the game is

Moderate

A

$\frac{m + 2}{2^{2 n} + 1}$
B

$\frac{2 n + 2}{2^{2 n}}$
C

$\frac{2 n + 2}{2^{2 n + 1}}$
D

$\frac{m + 2}{2^{2 n}}$

Solution

Player should get (HT, HT, HT,...) or (TH, TH,...) at least 2n times. If the sequence starts from first place, then the probability is 1/2²ⁿ and if starts from any other place, then the probability is $\frac{1}{2^{2 n + 1}}$ Hence, required probability is
$2 (\frac{1}{2^{2 n}} + \frac{m}{2^{2 n + 1}}) = \frac{m + 2}{2^{2 n}}$

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