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?

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

[tex] P_X(1) = {n \choose 1} * 0.5^1 * (1-0.5)^{n-1} = n * 0.5^n [/tex]

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.

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?

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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?