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Exam-style question
Use the skills together: This original question combines building a distribution, expected gain, a fair fee, expected losses and two independent plays.
A game uses two fair coins. The prize is 0 dollars for no heads, 4 dollars for one head and 8 dollars for two heads. A player pays 3 dollars to play.
(a) Find the probability distribution of the player’s net gain X. [3]
(b) Find E(X) and interpret your answer. [2]
(c) Find the entry fee that makes the game fair. [2]
(d) The game is played 60 times. Find the expected number of losses. [2]
(e) Two plays are independent. Find the probability that the total prize is 8 dollars. [3]
(a) What do you gain or lose?
You pay 3 dollars to play: X = prize − 3 dollars.
A negative X means you lose money.
| Coin result | Prize | X: prize − cost |
|---|---|---|
| TT: no heads | 0 dollars | 0 − 3 = −3 dollars |
| HT: one head | 4 dollars | 4 − 3 = 1 dollar |
| TH: one head | 4 dollars | 4 − 3 = 1 dollar |
| HH: two heads | 8 dollars | 8 − 3 = 5 dollars |
4 equally likely results: each has probability 1/4. HT and TH both give X = 1, so add their chances: 1/4 + 1/4 = 1/2.
| X: gain or loss | Results giving this X | Probability |
|---|---|---|
| −3 dollars | TT | |
| 1 dollar | HT or TH | |
| 5 dollars | HH |
(b) Average gain
Find E(X) and explain what it means.
Step by step
- (b) Multiply each net gain by its probability and add.
- Interpret: the player gains an average of 1 dollar per game in the long run.
Final answer
Over many games, the player gains an average of 1 dollar per game.
(c) A fair entry fee
What should the player pay so the game is fair?
Step by step
- The four equally likely results pay $0, $4, $4 and $8. Add the prizes and divide by 4.
- On average you receive $4. Pay $4 to leave no average gain or loss.
Final answer
Fair entry fee: $4.
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(d) Expected losses
Using the original 3-dollar entry fee, how many losses do we expect in 60 games?
Step by step
- You lose money only with TT: you pay 3 dollars and win nothing. TT is 1 of the 4 equally likely results.
- The chance of losing is 1/4. Take one quarter of 60 games.
Final answer
Expect 15 losses on average. A particular set of 60 games could have more or fewer.
(e) Total prize from two plays
Find the chance that the two prizes add to 8 dollars.
Step by step
- There are three ways to make $8: $0 then $8, $4 then $4, or $8 then $0.
- For each pair, multiply the two chances. The plays are independent.
- Add the chances of these three different ways.
Final answer
Probability = 3/8 = 0.375.
Check what the question asks: A prize is before the entry fee. A gain is after the fee.
In part (e), the total is a prize total, so use prizes 0, 4 and 8.