## Solution to: Faites Vos Jeux

Let the original bet be equal to *b*.
The probability that you lose a round is ^{18}/_{37}.
The probability that you lose *n* consecutive rounds
(which means that you must quit with a loss) is
(^{18}/_{37})^{n}.
Therefore, the probability that you win once is 1-(^{18}/_{37})^{n}.
If you lose *n* consecutive rounds, your total bet (and loss) equals (2^{n}-1) × *b*.
If you win once, your profit equals *b*.
So, the expected value for your profit is (^{18}/_{37})^{n} × ( -(2^{n}-1) × *b* ) + ( 1-(^{18}/_{37})^{n} ) × *b* = ( 1-(^{38}/_{37})^{n} ) × *b*.

Note that this expected value is *negative* for all values of *n* greater than 0.
You cannot expect a profit, but a loss!

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