Expected value and fair games
You are at a stall at a school fundraiser. It costs €2 to spin a wheel and the prizes are €0, €2 or €5. Some people win, most people don't. Is it worth playing? Expected value gives you a way to answer this with a number, not a hunch.
Expected value
Imagine you play a game a very large number of times. Sometimes you win a lot, sometimes nothing. The expected value is the average amount you win per play in the long run.
Why does this work? Suppose you play 100 times and the chance of winning €5 is $0.2$. You expect about 20 wins of €5, which is €100 in total. Dividing by 100 plays gives €1 per play, which is $5 \times 0.2$.
A table keeps the working tidy.
A spinner at a school fundraiser in Ennis gives the following prizes with these probabilities: €0 with probability $0.5$, €2 with probability $0.3$ and €5 with…
A spinner at a school fundraiser in Ennis gives the following prizes with these probabilities: €0 with probability $0.5$, €2 with probability $0.3$ and €5 with probability $0.2$. Calculate the expected value of the prize in one spin.
A fair six-sided die is rolled once. Let $X$ be the number shown. a) Write down the probability of each outcome. b) Calculate the expected value of $X$.
A fair six-sided die is rolled once. Let $X$ be the number shown.
- a) Write down the probability of each outcome.
- b) Calculate the expected value of $X$.
Raffles and expected winnings
Many examples are raffles and prize draws. Here you find the probability of each prize first, then use the formula.
The outcome "win nothing" contributes $0 \times p = 0$, so you can leave it out of the sum. Still, write it in your table so the probabilities add to 1.
In a raffle in Tralee, 200 tickets are sold at €1 each. There is one prize of €100. Aoife buys one ticket. a) What is the probability that Aoife wins the prize? b)…
In a raffle in Tralee, 200 tickets are sold at €1 each. There is one prize of €100. Aoife buys one ticket.
- a) What is the probability that Aoife wins the prize?
- b) Calculate the expected value of the prize for Aoife's ticket.
Profit and fair games
The prize is not the whole story, because the player pays to enter. The expected profit takes the cost into account:
A game is fair if the expected profit is zero, that is, if the player pays exactly what they expect to win. If the expected profit is negative the game favours the organiser, and if it is positive it favours the player.
At a game stall in Galway, a player pays €3 to play. Two coins are tossed. If two heads appear the player wins €10, if exactly one head appears the player wins €2, and…
At a game stall in Galway, a player pays €3 to play. Two coins are tossed. If two heads appear the player wins €10, if exactly one head appears the player wins €2, and otherwise the player wins nothing.
- a) Write down the probability of each outcome.
- b) Calculate the expected profit of the player per game.
- c) Is the game fair? Explain.
A game in a Cork arcade costs €2 to play. A card is drawn at random from a standard deck of 52 cards. If it is an ace the player wins €15, if it is any other picture…
A game in a Cork arcade costs €2 to play. A card is drawn at random from a standard deck of 52 cards. If it is an ace the player wins €15, if it is any other picture card (jack, queen or king) the player wins €5, and otherwise the player wins nothing.
- a) Calculate the expected winnings from one play.
- b) Calculate the expected profit for the player.
- c) How much should the organiser charge to play so that the game is fair?
Finding unknowns and decision making
Expected value is used to make decisions: whether to play, whether to buy insurance, or what to charge. Sometimes one number in the problem is unknown, and you set the expected profit to $0$ (or to a target) and solve.
Without replacement, probabilities change after each draw. For example, with 3 red and 5 blue counters in a bag, the chance both of two counters drawn are red is $\tfrac38 \times \tfrac27 = \tfrac{6}{56} = \tfrac{3}{28}$. Work through the draws one at a time, then use the expected value formula as usual.
Expected value is about the long run. A customer who plays once can still win, even if the expected profit is negative. The expected value only tells you what happens on average over many plays. The organiser plays the game thousands of times, so for them the average is reliable.
Now we can answer the opening question. A wheel gives €0 with probability $0.5$, €2 with $0.3$ and €5 with $0.2$, and costs €2. Then $E = 0 + 0.6 + 1.0 = €1.60$ and the expected profit is $-€0.40$, so it is not worth playing for profit, though it raises money for the school.
A farmer in County Meath estimates that the profit on a crop depends on the weather: €12,000 with probability $0.5$ in good weather, €7,000 with probability $0.3$ in…
A farmer in County Meath estimates that the profit on a crop depends on the weather: €12,000 with probability $0.5$ in good weather, €7,000 with probability $0.3$ in average weather, and a loss of €3,000 with probability $0.2$ in poor weather.
- a) Calculate the expected profit.
- b) The farmer can buy insurance for €800 which guarantees a minimum profit of €7,000 (so poor weather gives €7,000 before the cost of insurance). Calculate the expected profit with insurance, including the cost, and state whether the insurance is worthwhile on expected value.
A game at a Dublin charity event is played as follows. A player pays €4 and rolls a fair die. If a 6 is rolled the player wins €12. If a 5 is rolled the player wins €4.…
A game at a Dublin charity event is played as follows. A player pays €4 and rolls a fair die. If a 6 is rolled the player wins €12. If a 5 is rolled the player wins €4. Otherwise the player wins nothing.
- a) Calculate the expected profit per game.
- b) The organiser changes the prize for a 6 to €$x$. Find $x$ so that the game is fair.
- c) Using the original rules, the organiser expects 300 people to play. Estimate the organiser's total profit and comment on your answer.
A bag contains 3 red counters and 5 blue counters. A player pays €$c$ to draw two counters without replacement. The player wins €10 if both are red, €3 if exactly one is…
A bag contains 3 red counters and 5 blue counters. A player pays €$c$ to draw two counters without replacement. The player wins €10 if both are red, €3 if exactly one is red, and nothing if both are blue.
- a) Show that the probability of drawing two red counters is $\frac{3}{28}$ and find the probabilities of the other two outcomes.
- b) Calculate the expected winnings.
- c) Find the value of $c$ for which the game is fair.
A company offers a scratch card costing €5. The prizes are: €100 with probability $0.01$, €20 with probability $0.04$, €5 with probability $0.15$ and nothing otherwise.…
A company offers a scratch card costing €5. The prizes are: €100 with probability $0.01$, €20 with probability $0.04$, €5 with probability $0.15$ and nothing otherwise.
- a) Calculate the expected profit per card for a customer.
- b) The company wants an expected profit of €1 per card for itself. Keeping the other prizes and probabilities fixed, find the new value of the top prize.
- c) Explain why a customer who buys a single card may still win even though the expected profit is negative.
Summary
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Expected value: $E(X) = \sum x\,p$. Multiply each value by its probability and add. The probabilities must total 1.
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It is a long-run average and need not be one of the outcomes.
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Expected profit = expected winnings − cost to play.
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A game is fair when the expected profit is $0$, so the fair price equals the expected winnings.
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For an unknown value, set the expected profit to $0$ (or to a target) and solve.
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Use expected value to compare choices, but remember that a single play can differ greatly from the average.