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Unit 11 · 2.7

Expected value

Outcome × probability, then add — with NET gain

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Rule cards

Expected value with NET gain

E = Σ (outcome × probability), where each outcome is the NET gain = prize − cost (a loss is −cost).

outcome × probability, then add (E(X) = Σ x · P(x))

  1. Shortcut for raffles: (total prize $ ÷ tickets) − ticket price.
  2. Grab bags: [15(0) + 6(−1) + 2(2) + 7 + 17] ÷ 25 = $0.88.
  3. Die game, pay $2: outcomes −2, −2, −1, 0, 1, 2 each 1/6 → E = −$0.33.
Traps
  • Used the prize without subtracting the ticket price.
    $1000 prize, 1000 tickets at $3 → 1000(1/1000) = $1
    ✓ 997(1/1000) + (−3)(999/1000) = −$2 (exam Q27)
  • Left out the losing tickets.
    only the winning rows
    ✓ losers: (−3)(999/1000)
  • Counted a multi-prize line as one prize.
    two $500 prizes → $500
    ✓ $1000 + 2×$500 + 10×$100 = $3000 → 3000/1000 − 3 = $0 (exam Q28)

Fair game and insurance

E = 0 → fair; E < 0 → you lose on average. Fair price: set E = 0 and solve.

  1. Doubles pay $5: (6/36)(5 − x) + (30/36)(−x) = 0 → x = $0.83.
  2. Pick-3 straight ($1, wins $500): −$0.50; boxed 1-2-3 wins on 6 orders → −$0.52.
Traps
  • Called a game fair because you can win.
    you might win $5 → fair
    ✓ fair means E = 0
  • Took the company's view when the player's was asked (or reverse).
    insurance → −$265.70 for the company
    ✓ company: premium − payout × P(claim) = 360 − 100,000(.000943) = $265.70

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