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Introduction to Probability · Ages 15–16

Calculating Expected Value and Fair Price Practice

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Designed for 10th-grade probability students. Complete each problem by finding the expected payoff or net gain. Show your multiplication steps.

  1. A spinner has 4 equal sections labeled $2, $4, $8, and $10. What is the expected payout for one spin?

  2. You roll a standard 6-sided die. Rolling a 6 pays $12, while any other number pays $0. Calculate the expected payout.

  3. A bag contains 3 red marbles and 7 blue marbles. Drawing a red marble pays $30; drawing a blue marble pays $0. What is the expected payout?

  4. A raffle sells 200 tickets at $5 each for one prize of $400. What is the expected net value of purchasing one ticket?

  5. A game pays $20 with probability 0.1, $5 with probability 0.4, and $0 with probability 0.5. What entry fee makes this a fair game?

  6. A game costs $2 to play. You flip two coins. If both are heads, you win $6; otherwise, you win $0. Is this game fair, favorable to the player, or unfavorable?

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Calculating Expected Value and Fair Price Practice

Answer key

  1. A spinner has 4 equal sections labeled $2, $4, $8, and $10. What is the expected payout for one spin?

    $6.00. Multiply each value by 0.25: ($2 · 0.25) + ($4 · 0.25) + ($8 · 0.25) + ($10 · 0.25) = $0.50 + $1.00 + $2.00 + $2.50 = $6.00.

  2. You roll a standard 6-sided die. Rolling a 6 pays $12, while any other number pays $0. Calculate the expected payout.

    $2.00. The probability of rolling a 6 is 1/6, and any other number is 5/6. ($12 · 1/6) + ($0 · 5/6) = $2.00.

  3. A bag contains 3 red marbles and 7 blue marbles. Drawing a red marble pays $30; drawing a blue marble pays $0. What is the expected payout?

    $9.00. P(Red) = 3/10 and P(Blue) = 7/10. ($30 · 0.3) + ($0 · 0.7) = $9.00.

  4. A raffle sells 200 tickets at $5 each for one prize of $400. What is the expected net value of purchasing one ticket?

    -$3.00. The gross expected value is $400 · (1/200) = $2.00. Subtract the $5.00 ticket price to get a net expected value of -$3.00.

  5. A game pays $20 with probability 0.1, $5 with probability 0.4, and $0 with probability 0.5. What entry fee makes this a fair game?

    $4.00. Calculate expected gross payout: ($20 · 0.1) + ($5 · 0.4) + ($0 · 0.5) = $2.00 + $2.00 + $0 = $4.00. Charging $4.00 makes net expected value zero.

  6. A game costs $2 to play. You flip two coins. If both are heads, you win $6; otherwise, you win $0. Is this game fair, favorable to the player, or unfavorable?

    Unfavorable. P(HH) = 1/4 = 0.25. The expected payout is $6 · 0.25 = $1.50. Net expected value is $1.50 - $2.00 = -$0.50, so it is unfavorable to the player.