Data and probability · Ages 14–15

Fair games and expected gain worksheet

Part of: Probability

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Data and probability · Ages 14–15

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.One play costs 6 dollars. A die is rolled: a 6 wins 6 dollars, anything else nothing. What is the expected gain from one play, in dollars (a loss as a negative number)?
Answer:
2.One play costs 6 dollars. A die is rolled: a 6 wins 30 dollars, anything else nothing. What is the expected gain from one play, in dollars (a loss as a negative number)?
Answer:
3.A scratch card costs 4 dollars and has a 1100 chance of winning 400 dollars, otherwise nothing. The expected prize is dollars; the expected gain on one card is dollars (a loss as a negative number).
Fill in each blank
Answer:
4.A stall's spinner has 5 equal parts, each number a prize in dollars, as shown. For the game to be fair, how many dollars should one play cost?
Answer:
5.A scratch card costs 8 dollars and has a 120 chance of winning 200 dollars, otherwise nothing. The expected prize is dollars; the expected gain on one card is dollars (a loss as a negative number).
Fill in each blank
Answer:
6.One play costs 2 dollars. A die is rolled: a 5 wins 5 dollars, a 6 wins x dollars, anything else nothing. For the game to be fair, x = .
Answer:
7.A stall game: a bag holds 2 red, 4 blue and 4 white balls, and one is drawn at random. Red wins 22 dollars, blue 18 dollars, white nothing. For the game to be fair, how many dollars should one play cost?
Answer:
8.One play costs 5 dollars. A die is rolled: a 5 wins 8 dollars, a 6 wins x dollars, anything else nothing. For the game to be fair, x = .
Answer:

9.One play costs 4 dollars; rolling a 6 wins 24 dollars, anything else nothing. Which description is right?
- A. On average it favours the player
- B. It can't be told
- C. On average it favours the stall
- D. The game is fair
Choose the right answer
Answer:
10.One play costs 7 dollars; rolling a 6 wins 24 dollars, anything else nothing. Which description is right?
- A. On average it favours the stall
- B. On average it favours the player
- C. The game is fair
- D. It can't be told
Choose the right answer
Answer:
11.A coin is tossed: heads wins 10 dollars, tails loses 11 dollars. The game is fair.
True or false?
Answer:
12.A coin is tossed: heads wins 2 dollars, tails loses 2 dollars. The game is fair.
True or false?
Answer:

13.An insurance policy costs 750 dollars a year. The chance of a claim of 14600 dollars in a year is 1200. What is the insurer's expected gain per policy per year, in dollars?
Working:
Answer:
14.May pays 4 dollars for a raffle ticket. There are 200 tickets and one wins 1400 dollars. What is May's expected gain, in dollars (a loss as a negative number)?
Working:
Answer:
15.Tsz-hin's stall game uses a spinner of 4 equal parts, each number a prize in dollars, as shown. One play costs 5 dollars. The expected prize is dollars; on average the stall makes dollars a play.
Fill in each blank
Working:
Answer:
16.Game A: roll a die, and a 6 wins 54 dollars. Game B: toss a coin, and heads wins 16 dollars. Both are free, and other outcomes win nothing. Game A's expected prize is dollars, and game B's .
Fill in each blank
Working:
Answer:
17.A stall game rolls a die: a 1 wins 1 dollars, a 6 wins 41 dollars, anything else nothing. For the game to be fair, what should one play cost?
- A. 8 dollars
- B. 7 dollars
- C. 41 dollars
- D. 21 dollars
Choose the right answer
Working:
Answer:
18.Ka-yan pays 5 dollars for a raffle ticket. There are 100 tickets and one wins 200 dollars. What is Ka-yan's expected gain, in dollars (a loss as a negative number)?
Working:
Answer:

Show the answers and working

Fair games and expected gain worksheet: Answers and working

1. −5 dollars
Expected prize = 6 × 16 = 1; expected gain = 1 − 6 = −5.
2. −1 dollars
Expected prize = 30 × 16 = 5; expected gain = 5 − 6 = −1.
3. 4, 0
Expected prize = 400 × 1100 = 4; expected gain = 4 − 4 = 0.
4. 14 dollars
Fair means price = expected prize = 70 ÷ 5 = 14 dollars.
5. 10, 2
Expected prize = 200 × 120 = 10; expected gain = 10 − 8 = 2.
6. 7
Fair: 5 × 16 + x × 16 = 2, so 5 + x = 12 and x = 7.
7. 11.6 dollars
10 balls: P(red) = 210, P(blue) = 410. Fair means price = expected prize = 22 × 210 + 18 × 410 = 116 ÷ 10 = 11.6 dollars.
8. 22
Fair: 8 × 16 + x × 16 = 5, so 8 + x = 30 and x = 22.

9. D. The game is fair
Expected prize = 24 × 16 = 4 dollars against a price of 4: equal is fair, more favours the player, less the stall.
10. A. On average it favours the stall
Expected prize = 24 × 16 = 4 dollars against a price of 7: equal is fair, more favours the player, less the stall.
11. False
Expected gain = 10 × 12 − 11 × 12, which is 0 only when the win and the loss are equal.
12. True
Expected gain = 2 × 12 − 2 × 12, which is 0 only when the win and the loss are equal.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 677 dollars
Expected payout = 14600 × 1200 = 73; expected gain = 750 − 73 = 677 dollars.
14. 3 dollars
Expected prize = 1400 × 1200 = 7; expected gain = 7 − 4 = 3.
15. 3.75, 1.25
Expected prize = 15 ÷ 4 = 3.75; the stall makes 5 − 3.75 = 1.25 dollars a play on average.
16. 9, 8
A: 54 × 16 = 9; B: 16 × 12 = 8. The higher one pays more on average over many plays.
17. B. 7 dollars
Fair means price = expected prize = (1 + 41) × 16 = 7 dollars.
18. −3 dollars
Expected prize = 200 × 1100 = 2; expected gain = 2 − 5 = −3.

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