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In a $2-to-play game, a computer draws five digits with replacement, and matching all five in order pays $100,000 plus your $2 back. The probability of winning is 0.00001. What does the expected-value table say about your long-run result from this game?
AYou win about $1 per game, on average
BYou break even over the long run
CYou lose about $1 per game, on average
DYou lose about $100,000 per game, on average
Answer & Solution
Correct answer: C. You lose about $1 per game, on average
1. Let X be your profit: a loss of $2 with probability 0.99999, or a profit of $100,000 with probability 0.00001.
2. Multiply and add: (-2)(0.99999) + (100000)(0.00001) = -1.99998 + 1 = -0.99998.
3. This expected value is about -$1, so on average you lose roughly one dollar per game.
4. Option A reverses the sign, and option D mistakes the rare $100,000 prize for the typical outcome, so both are wrong.
_Source: OpenStax Introductory Statistics (CC BY 4.0), Ch 4 "Discrete Random Variables", section 4.2 Mean or Expected Value and Standard Deviation_
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