1963 AMC 12 Problem 36

Question

A person starting with

64

cents and making

6

bets, wins three times and loses three times, the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. If each wager is for half the money remaining at the time of the bet, then the final result is:

a loss of

27

¢

a gain of

27

¢

a loss of

37

¢

neither a gain nor a loss

a gain or a loss depending upon the order in which the wins and losses occur

Answer: C

Concepts:exponent process simulation

Difficulty rating: 1470

Small hint:

A win multiplies the current amount by

\frac{3}{2}

, while a loss multiplies it by

\frac{1}{2}

Big hint:

Multiplication makes the order irrelevant

Solution:

After three wins and three losses, the amount, in cents, is

\begin{aligned} 64\left(\frac32\right)^3 \left(\frac12\right)^3 &=64\cdot\frac{27}{64}\\ &=27. \end{aligned}

The loss is

64-27=37

cents, regardless of order.

Thus, the correct answer is C.