Question

Teri designed a game to play against her brother Will in which Teri flips a fair coin n times, and Will wins if and only if exactly one of the flips lands heads up.

If there is a 25% chance of Will winning the game, what is the value of n?

Answer 1

4

Explanation:

Since the coin is fair, then the probability of each flip to be heads is 0.5. The number of heads obtained from flipping n coins is a random variable, that we will note X, with Binomial distribution B(n,0.5).

The probability of X being equal to 1 is

`P_X(1) = {n \choose 1} * 0.5^1 * (1-0.5)^(n-1) = n * 0.5^n`

If n = 4, then PX(1) = 4 * 0.5⁴ = 0.25.

We can chech for small values:

• For n = 1: 1 * 0.5 = 0.5
• for n = 2: 2*0.5² = 0.5
• for n=3: 3*0.5³ = 3/8 = 0.375
• for n=5: 5*0.5⁵ = 5/32 = 0.15625

The probability decreaces as long as n increase. So the only value of n for which the probability is 0.25 is n=4.