In a game, the probability of scoring 0 is 0.1, of scoring 8 is 0.6, and of scoring 11 is 0.3. The expected score is □.

8.1. E = 0 × 0.1 + 8 × 0.6 + 11 × 0.3 = 8.1.

A fair die is rolled: a 6 wins 35 dollars, a 1 wins 1 dollars, anything else nothing. The probability of a 6 is □, and the expected prize □ dollars. Fill in each blank

1/6, 6. E = 35 × 1/6 + 1 × 1/6 + 0 × 4/6 = (35 + 1) ÷ 6 = 6.

A spinner has 7 equal parts numbered 1 to 7. Spun once, what is the expected number? A. 3.5 B. 7 C. 8 D. 4 Choose the right answer

D. 4. E = (1 + 2 + … + 7) × 1/7 = (1 + 7) ÷ 2.

Probability · Ages 14–15

Expected value

E = Σ value × probability: the average over many tries.

How to do it

Method

E = Σ value × probability: the average over many tries

Spinner $0, $0, $10, $30: E = 40 × 14 = $10

Another way

Each outcome once, then the mean

(0 + 0 + 10 + 30) ÷ 4 = 10

In full

An expected value is what you get on average each time over very many tries: E = Σ (each value × its probability). For a fair die, E = 1 × 16 + 2 × 16 + … + 6 × 16 = 216 = 3.5. It needn't be one of the outcomes: one roll never gives 3.5, but over many rolls the average is about 3.5. A spinner of 4 equal parts with prizes $0, $0, $10, $30: E = 0 × 14 + 0 × 14 + 10 × 14 + 30 × 14 = $10. Another way: picture each outcome happening exactly once and take the mean: (0 + 0 + 10 + 30) ÷ 4 = $10, the same. An expected number = trials × probability: 60 rolls of a die should give a 6 about 60 × 16 = 10 times.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
In a game, the probability of scoring 0 is 0.1, of scoring 8 is 0.6, and of scoring 11 is 0.3. The expected score is .
Show the answer and working
Answer: 8.1
E = 0 × 0.1 + 8 × 0.6 + 11 × 0.3 = 8.1.
Example 2
A fair die is rolled: a 6 wins 35 dollars, a 1 wins 1 dollars, anything else nothing. The probability of a 6 is , and the expected prize dollars.
Fill in each blank
Show the answer and working
Answer: 16, 6
E = 35 × 16 + 1 × 16 + 0 × 46 = (35 + 1) ÷ 6 = 6.
Example 3
A spinner has 7 equal parts numbered 1 to 7. Spun once, what is the expected number?
- A. 3.5
- B. 7
- C. 8
- D. 4
Choose the right answer
Show the answer and working
Answer: D. 4
E = (1 + 2 + … + 7) × 17 = (1 + 7) ÷ 2.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.A game's spinner has 8 equal parts, each number a score, as shown. Spun once, the expected score is .
Answer:
2.A fair die is rolled 78 times. A 6 or more should come up about times.
Answer:
3.A fair die is rolled: a 6 wins 31 dollars, a 1 wins 17 dollars, anything else nothing. The probability of a 6 is , and the expected prize dollars.
Fill in each blank
Answer:
4.In a game, the probability of scoring 0 is 0.3, of scoring 4 is 0.4, and of scoring 44 is 0.3. The expected score is .
Answer:
5.In a game, scoring 17 and scoring 1 each have probability 12. The expected score is 9.
True or false?
Answer:

All the answers

1. 20.625
2. 13
3. 16, 8
4. 14.8
5. True

Free worksheet: Expected valueWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

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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