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← Unit 11 · Expected value

Lesson

Read the rule, then work each example: try the setup yourself, then tap to check one step at a time.

The rules

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.

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 and calculator keys are on the unit page →

Worked examples (5)

Example 1 · 2.7 #1a0/5

Find the expected value if one ticket is purchased

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One hundred raffle tickets are sold for $1 each. One prize of $50 will be awarded.

Try the setup yourself first, then tap.

Example 2 · 2.7 #1b0/5

Assume that there are three prizes to be awarded. One prize of $30 and two prizes of 10 are awarded, find the expected value if one ticket is purchased.

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One hundred raffle tickets are sold for $1 each. One prize of $50 will be awarded.

Try the setup yourself first, then tap.

Example 3 · 2.7 #2a0/5

Find the expected value

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An outdoor concert featuring a very popular musical group is scheduled for a Sunday afternoon in a large open stadium. The promoter, worrying about being rained out, contacts a long-range weather forecaster who predicts the chance of rain on that Sunday to be .24. If it does not rain, the promoter is certain to net $100,000; if it does rain, the promoter estimates that the net will be only $10,000.

Try the setup yourself first, then tap.

Example 4 · 2.7 #2b0/5

If an insurance company agrees to insure the concert for $100,000 against rain at a premium of $20,000, should the promoter buy the insurance?

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An outdoor concert featuring a very popular musical group is scheduled for a Sunday afternoon in a large open stadium. The promoter, worrying about being rained out, contacts a long-range weather forecaster who predicts the chance of rain on that Sunday to be .24. If it does not rain, the promoter is certain to net $100,000; if it does rain, the promoter estimates that the net will be only $10,000.

Try the setup yourself first, then tap.

Example 5 · 2.7 #2c0/5

Find the expected value from the insurance point of view

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An outdoor concert featuring a very popular musical group is scheduled for a Sunday afternoon in a large open stadium. The promoter, worrying about being rained out, contacts a long-range weather forecaster who predicts the chance of rain on that Sunday to be .24. If it does not rain, the promoter is certain to net $100,000; if it does rain, the promoter estimates that the net will be only $10,000.

Try the setup yourself first, then tap.

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