Probability · Ages 14–15
Fair games and expected gain
Expected gain = expected prize − cost; 0 is fair.
How to do it
Method
Expected gain = expected prize − cost; 0 is fair
24 × 16 − 5 = 4 − 5 = −1
Another way
Picture 6 plays
(24 − 6 × 5) ÷ 6 = −1
In full
The expected gain from one play = the expected prize − the cost. An expected gain of 0 makes a game fair; above 0 it favours the player on average; below 0 the player loses on average each play, and it favours the stall. $5 a play, a prize of $24 for rolling a 6: the expected prize = 24 × 16 = $4, so the expected gain = 4 − 5 = −$1, a loss of $1 a play on average. To make it fair, the price should be the expected prize, $4. Another way: picture 6 plays, each outcome once: $30 paid, $24 won, an average of (24 − 30) ÷ 6 = −$1 a play, the same. A negative expected gain doesn't mean losing every time, only losing on average over many plays.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
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?
Show the answer and working
Answer: 12 dollars
Fair means price = expected prize = 60 ÷ 5 = 12 dollars.
Example 2
A coin is tossed: heads wins 9 dollars, tails loses 8 dollars. The game is fair.
True or false?
Show the answer and working
Answer: False
Expected gain = 9 × 12 − 8 × 12, which is 0 only when the win and the loss are equal.
Example 3
One play costs 2 dollars. A die is rolled: a 5 wins 2 dollars, a 6 wins x dollars, anything else nothing. For the game to be fair, x = .
Show the answer and working
Answer: 10
Fair: 2 × 16 + x × 16 = 2, so 2 + x = 12 and x = 10.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.One play costs 6 dollars; rolling a 6 wins 30 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:
2.One play costs 4 dollars. A die is rolled: a 6 wins 18 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 9 dollars and has a 110 chance of winning 60 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 8 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.One play costs 5 dollars. A die is rolled: a 5 wins 10 dollars, a 6 wins x dollars, anything else nothing. For the game to be fair, x = .
Answer:
All the answers
- 1. C. On average it favours the stall
- 2. −1 dollars
- 3. 6, −3
- 4. 17.5 dollars
- 5. 20
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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