Question 1110074
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A fair game means you break even after several games.  Just looking at the numbers, it takes 2 wins at $4 each to balance 1 loss at -$8.  So you need to win 2 out of every 3 games to break even.  That makes the probability of winning 2/3.<br>
Formally, you want the expected value of a game to be 0.  If the probability of winning is x, then the probability of losing is (1-x); for the expected value to be 0, we need to have
{{{4(x)+(-8)(1-x) = 0}}}
{{{4x-8+8x = 0}}}
{{{12x-8 = 0}}}
{{{12x = 8}}}
{{{x = 2/3}}}