Worksheets · Foundation and Higher

Exhaustive outcomes and experimental probability

8 exam-style questions, grades 1 to 6. Worked solutions and the marks are on the last page.

Question 1Non-calculator · 2 marks
(a) A machine selects one of three symbols: a star, a circle or a square. The probability of a star is 0.28 and the probability of a circle is 0.47. What is the probability of a square? (2)
1. 0.19
2. 0.25
3. 0.75
4. 1.25
Question 2Non-calculator · 2 marks
(a) A drawing pin is dropped 150 times. It lands point up 57 times. Use these results to estimate the probability that it lands point down on the next drop. Give your answer as a decimal. (2)
Question 3Non-calculator · 3 marks
A bag holds red, blue, green and yellow counters. A counter is taken at random. P(red) $= 0.2$ , P(blue) $= 0.35$ , P(yellow) $= 0.15$ .
(a) Work out the probability that the counter is green. (2)
(b) Work out the probability that the counter is not blue. (1)
Question 4Non-calculator · 3 marks
A spinner numbered 1, 2 and 3 is spun 60 times. It lands on 1 twelve times, on 2 eighteen times and on 3 thirty times.
(a) Work out the relative frequency of the spinner landing on 3. (1)
(b) The spinner is spun 300 more times. Work out an estimate for the number of times it lands on 3. (2)
Question 5Non-calculator · 4 marks
The only possible outcomes of a game are win, draw and lose. The probabilities of win and draw are $3/20$ and $1/5$ .
(a) Find the probability of losing. (2)
(b) The game is played 200 times. Find the expected number of losses. (2)
Question 6Non-calculator · 3 marks
A machine is tested 150 times. It succeeds 51 times.
(a) Estimate the failure probability. (2)
(b) Why would a much larger test under the same conditions usually improve this estimate? (1)
Question 7Non-calculator · 4 marks
Ella throws a coin 50 times and gets 31 heads. Tom throws the same coin 500 times and gets 262 heads.
(a) Work out the relative frequency of heads in Ella's throws. (1)
(b) Whose results give the better estimate of the probability of heads? Give a reason. (1)
(c) Use the better estimate to work out how many heads to expect in 1000 throws. (2)
Question 8Non-calculator · 4 marks
A spinner can land on $A$ , $B$ , $C$ or $D$ . P( $A$ ) $= 2x$ , P( $B$ ) $= x + 0.1$ , P( $C$ ) $= 0.3$ and P( $D$ ) $= x$ .
(a) Work out P( $B$ ). (3)
(b) The spinner is spun 200 times. Work out an estimate for the number of times it lands on $B$ . (1)

Worked solutions and marks

Question 1

(a) 0.25

$1-0.28-0.47$
Probabilities of all possible outcomes add to 1.
P(square) = 1 $-$ (0.28 + 0.47) = 0.25.

P1 Establishing $1-0.28-0.47$ or an equivalent valid method.
A1 Correct answer: 0.25

Question 2

(a) $0.62$

$\frac{150-57}{150}$
Point down occurred 150 $-$ 57 = 93 times.
Relative frequency estimates probability: 93/150 = 0.62.

P1 Establishing $\frac{150-57}{150}$ or an equivalent valid method.
A1 Correct answer: $0.62$

Question 3

(a) 0.3

The four colours are all the possible outcomes, so the probabilities add up to 1.
$1 - (0.2 + 0.35 + 0.15) = 1 - 0.7 = 0.3$

M1 $1 - (0.2 + 0.35 + 0.15)$ .
A1 Correct answer: 0.3.

(b) 0.65

$1 - 0.35 = 0.65$ .

B1 Correct answer: 0.65.

Question 4

(a) 0.5

$\frac{30}{60} = 0.5$ .

B1 0.5 or equivalent.

(b) 150

$0.5 \times 300 = 150$ .

M1 Relative frequency times 300.
A1 Correct answer: 150.

Question 5

(a) $13/20$

Exhaustive mutually exclusive outcomes have total probability 1.
$1-(3/20)-(1/5)$
Therefore $13/20$ .

M1 Exhaustive mutually exclusive outcomes have total probability 1.
A1 Correct answer: $13/20$

(b) $130$

Multiply the loss probability by the number of games.
$200\times (13/20)$
Therefore $130$ .

M1 Multiply the loss probability by the number of games.
A1 Correct answer: $130$

Question 6

(a) $33/50$

Find failures as a fraction of all tests.
$\frac{150-51}{150}$
Therefore $33/50$ .

M1 Find failures as a fraction of all tests.
A1 Correct answer: $33/50$

(b) Random fluctuations tend to have less effect on the relative frequency in a larger representative sample.

Random fluctuations tend to have less effect on the relative frequency in a larger representative sample.

C1 Correct conclusion with supporting reasoning: Random fluctuations tend to have less effect on the relative frequency in a larger representative sample.

Question 7

(a) 0.62

Relative frequency $=$ number of heads $\div$ number of throws $= \frac{31}{50} = 0.62$ .

B1 0.62 or $\frac{31}{50}$ .

(b) Tom's, because he made more throws.

Relative frequency tends to settle towards the true probability as the number of trials grows.

C1 Tom's, with a reason about the larger number of trials.

(c) 524

Tom's estimate: $\frac{262}{500} = 0.524$ .
$0.524 \times 1000 = 524$

M1 $\frac{262}{500} \times 1000$ .
A1 Correct answer: 524.

Question 8

(a) 0.25

The outcomes are exhaustive and mutually exclusive, so the probabilities add up to 1.
$2x + x + 0.1 + 0.3 + x = 1$
$4x + 0.4 = 1 \;\Rightarrow\; x = 0.15$
P( $B$ ) $= 0.15 + 0.1 = 0.25$ .

P1 Setting the sum of the four probabilities equal to 1.
P1 Solving to $x = 0.15$ .
A1 Correct answer: 0.25.

(b) 50

$0.25 \times 200 = 50$ .

B1 Correct answer: 50.

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