Infinite Rolling of Coins

Probability Level 4

Let p p be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 5 heads before one encounters a run of 2 2 tails. Given that p p can be written in the form m n , \frac{m}{n}, where m m and n n are relatively prime positive integers, find m + n . m + n.


The answer is 37.

Alan Yan by Alan Yan , 20, USA

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1 solution

Alan Yan
Nov 4, 2015

A = desired probability with coin toss starting with heads B = desired probability with coin toss starting with tails \begin{aligned} A & = \text{desired probability with coin toss starting with heads} \\ B & = \text{desired probability with coin toss starting with tails} \end{aligned} Some valid starts are: : T : H T : H H T : H H H T : H H H H T : H H H H H \begin{aligned} & : T \\ & : HT \\ & : HHT\\ & : HHHT\\ & : HHHHT\\ & : HHHHH \end{aligned}

Thus, A = B ( 1 2 + 1 4 + 1 8 + 1 16 ) + 1 32 B = 1 2 A A = 15 32 A + 1 32 ⟹ A = 1 17 , B = 1 34 A + B = 3 34 ⟹ m + n = 37 \begin{aligned} A & = B\left(\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}\right) + \frac{1}{32} \\ B & = \frac{1}{2}A \\ A & = \frac{15}{32}A + \frac{1}{32} \implies A = \frac{1}{17} , B = \frac{1}{34} \\ A + B & = \frac{3}{34} \implies m + n = \boxed{37} \end{aligned}

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Can you please provide some other solution

Madhulika Malhotra - 5 years, 7 months ago

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this is the standard method, but I will try.

Alan Yan - 5 years, 7 months ago

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