Question

Suppose you play a series of 100 independent games. If you win a game, you win 4 dollars. If you lose a game, you lose 4 dollars. The chances of winning each game is 1/2. Use the central limit theorem to estimate the chances that you will win more than 50 dollars.

Answer 1

You play a series of 100 independet games.

if you win a game, you win 4 dollars. If you lose a game, you lose 4 dollars.

The chances of winning each game is 1/2. So, the losing each game is 1/2.

So here,

P(Win) = P(Lose) = 0.50

Expected number of dollars won in a game = 4 * 1/2 - 4 * 1/2 = 0

Standard deviation of number of dollars win a single game = sqrt [(4 - 0)² * 1/2 + (0 + 4)² * 1/2] = 4

now for 100 games,

Expected number of dollars won in a 100 games = 100* ( 4 * 1/2 - 4 * 1/2) = 0

Standard deviation of number of dollars win a single game = sqrt (100 * 4) = 20

so here as the sample size is large we will employ central limit theorem and if x is the amount of winnings in 100 games. then,

x ~ N(0, 20)

we have to find

P(x > 50) = 1 - P(x < 50) = 1 - NORMSIDST(x < 50, 0, 20)

Z = (50 - 0)/20 = 2.5

P(x > 50) = 1 - P(z < 2.5) = 1 - 0.9938 = 0.0062