Don't gamble!

Probability Level 3

A gambler has 1 dollar in his pocket. He tosses an unbiased coin unless he has lost all his money or unless the coin is tossed for 5 times. For each head he wins a dollar, and for each tail he loses a dollar. Find the probability that the gambler will end up with $0.

If the answer is in the form $\frac{a}{b}$ , where $a$ and $b$ are coprime positive integers, find $a+b$ .

The answer is 27.

1 solution

Brian Charlesworth
Apr 2, 2017

The gambler's winnings must go up or down 1 dollar after each coin toss unless they have lost all their money. The only possible sequences of dollars on hand that result in them losing all their money are

$(0), (2,1,0), (2,1,2,1,0)$ and $(2,3,2,1,0)$ .

Bottoming out in a sequence of $n$ coin tosses occurs with probability $\dfrac{1}{2^{n}}$ , so as there is 1 losing sequence involving 1 toss, 1 involving 3 tosses and 2 involving 5 tosses, the probability that the gambler will lose all their money is

$\dfrac{1}{2} + \dfrac{1}{2^{3}} + 2 \times \dfrac{1}{2^{5}} = \dfrac{1}{2} + \dfrac{1}{8} + \dfrac{2}{32} = \dfrac{8 + 2 + 1}{16} = \dfrac{11}{16}$ , and thus $a + b = 11 + 16 = \boxed{27}$ .

Comment: The probabilities $P(k)$ of possessing $k$ dollars after 5 tosses are

$P(0) = \dfrac{11}{16}, P(1) = P(3) = P(5) = 0, P(2) = \dfrac{5}{32}, P(4) = \dfrac{1}{8}, P(6) = \dfrac{1}{32}$ .

Follow-up question: If this game were to continue for potentially an infinite number of tosses, what is the probability that the gambler will eventually lose all their money?

Could you give a hint to follow up question ?

Vishal Yadav - 4 years, 1 month ago

Follow-up question is a one-dimension random walk question. The gambler will lost all his money eventually, but the expectation of times of tosses does not exist.

Alstream Yang - 2 years, 3 months ago

Don't gamble!

The answer is 27.

1 solution

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