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In a coin game, P(heads) = 2/3 and P(tails) = 1/3. Tossing heads costs you $6 and tossing tails wins you $10. What does the expected value μ tell you about this game?
AYou win about 67 cents per game, on average
BYou lose exactly $6 every single time
CThe game is exactly fair, with μ = 0
DYou lose about 67 cents per game, on average
Answer & Solution
Correct answer: D. You lose about 67 cents per game, on average
1. Build the expected value table: x*P(x) for heads is (-6)(2/3) = -4, and for tails is (10)(1/3) = 10/3.
2. Add the column: -4 + 10/3 = -2/3, so μ = -2/3 of a dollar.
3. -2/3 of a dollar is about -$0.67, meaning an average loss of roughly 67 cents per game.
4. Option C wrongly claims the game is fair, but a nonzero μ shows it favors the house, not the player.
_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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