1963 AHSME Problems/Problem 36

Problem

A person starting with $$64$ 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:

$\textbf{(A)}\text{ a loss of }$ 27 \qquad \textbf{(B)}\text{ a gain of }$ 27 \qquad \textbf{(C)}\text{ a loss of }$ 37 \qquad \\ \textbf{(D)}\text{ neither a gain nor a loss}\qquad \\ \textbf{(E)}\text{ a gain or a loss depending upon the order in which the wins and losses occur}$

Solution

If the person wins the bet, the person has $\frac{3}{2}$ of the previous amount. If the person loses the bet, the person only has $\frac{1}{2}$ of the previous amount.

Because of the Commutative Property, the order of multiplying the multipliers does not matter. Thus, the person walks away with $64 \cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = 27$ dollars, so the person loses $$ 37$ . The answer is $\boxed{\textbf{(C)}}$ .

1963 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 35	Followed by Problem 37
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
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1963 AHSME Problems/Problem 36

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