Exercises: Expected value and fair games

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.

1. a) Write down the probability of each outcome.
2. b) Calculate the expected value of $X$.

In a raffle in Tralee, 200 tickets are sold at €1 each. There is one prize of €100. Aoife buys one ticket.

1. a) What is the probability that Aoife wins the prize?
2. b) Calculate the expected value of the prize for Aoife's ticket.

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.

1. a) Write down the probability of each outcome.
2. b) Calculate the expected profit of the player per game.
3. 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 card (jack, queen or king) the player wins €5, and otherwise the player wins nothing.

1. a) Calculate the expected winnings from one play.
2. b) Calculate the expected profit for the player.
3. c) How much should the organiser charge to play so that the game is fair?

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.

1. a) Calculate the expected profit.
2. 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. Otherwise the player wins nothing.

1. a) Calculate the expected profit per game.
2. b) The organiser changes the prize for a 6 to €$x$. Find $x$ so that the game is fair.
3. 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 red, and nothing if both are blue.

1. a) Show that the probability of drawing two red counters is $\frac{3}{28}$ and find the probabilities of the other two outcomes.
2. b) Calculate the expected winnings.
3. 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.

1. a) Calculate the expected profit per card for a customer.
2. 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.
3. c) Explain why a customer who buys a single card may still win even though the expected profit is negative.