Select Page

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 repeatedly flipping a fair coin.

## Flipping a Fair Coin – AIME I, 1995

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

• is 107
• is 37
• is 840
• cannot be determined from the given information

### Key Concepts

Integers

Probability

Algebra

But try the problem first…

Source

AIME I, 1995, Question 15

Elementary Number Theory by David Burton

## Try with Hints

First hint

B be tail flipped

outcomes are AAAAA, BAAAAA, BB. ABB, AABB, AAABB, AAAABB

Second Hint

with probabilities $\frac{1}{32}$, $\frac{1}{64}$, $\frac{1}{4}$, $\frac{1}{8}$, $\frac{1}{16}$, $\frac{1}{32}$, $\frac{1}{64}$

Final Step

with five heads AAAAA, BAAAAA sum =$\frac{3}{64}$ and sum of outcomes=$\frac{34}{64}$

or, m=3, n=34

or, m+n=37.