Expected Value and Fair Games

What is the expected value in a lottery game?

• The average amount of money a player can expect to win or lose in a single ticket

• The probability of winning the lottery

• The number of balls in a lottery machine

• The value of a lottery ticket

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How is expected value calculated from probability tables?

• By multiplying each outcome by its probability and summing the results

• By dividing the total number of outcomes by the probability of a specific outcome

• By subtracting the probability of a specific outcome from the total number of outcomes

• By adding the probability of a specific outcome to the total number of outcomes

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What is a fair game?

• A game where the expected value is zero

• A game where all players have an equal chance of winning

• A game where the probability of winning is high

• A game where the stakes are high

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In a lottery game, what does it mean to have a negative expected value?

• The player can expect to lose more money than they win in the long run

• The player has a low probability of winning the current ticket

• The player has a weak ticket

• The player has a small number of tickets

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Given the probability model in the table below, what is the expected value of the random variable?
  X        50          20        5
P(X)     0.1         0.3      0.6

• 14
• 4.67
• 5
• 7.5

Answer explanation

  X        50          20        5
P(X)     0.1         0.3      0.6

Multiply each outcome by the probability & add them up:

EV = 50(0.1) + 20(0.3) + 5(0.6) = 14

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Find the mean (expected value) of the probability distribution.

• 85

• 83.2

• 87.1

• 84.4

Answer explanation

Multiply each outcome by the probability & add them up:

EV = 75(.12) + 80(.23) + 85(.42) + 90(.11) + 95(.12)

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Eva flips a coin. If she gets heads, she wins $4. If she gets tails, she loses $3. What is her expected value of a coin flip?

• $1

• $0.50

• -$0.50

• $0

Answer explanation

P(H) = 0.5 and P(T) = 0.5

Multiply each outcome by the probability & add them up:

EV = $4(0.5) + -$3(0.5)

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What is the expected value of the spinner?

• 6.25

• 6.5

• 4.75

• 13

Answer explanation

P(-3) = 0.25 and P(10) = 0.25 and P(6) = 0.5

Multiply each outcome by it's probability and add them up:

EV = -3(0.25) + 10(0.25) + 6(0.5)

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If the expected value of this spinner game is $4.75, what should the cost of a ticket to spin be to make this a FAIR GAME?

• $0

• $3

• $4.75

• $13

Answer explanation

A fair game has an expected value = 0.

If the current EV ≠ 0, then the ticket price should counter balance it and MAKE it = 0.

Here, the EV = $4.25. This is the average you can expect to win per spin if you were to do a bunch of spins. That means the game host is losing $. It's not fair to them!

To make this a fair game, the ticket should cost $4.25.

NOW, the EV = -4.25 + -3(0.25) +10(0.25) + 6(0.5) = 0

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What is the expected value?

• -3.7

• 5.9

• 1.44

• -1.1

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