Two fair coins are tossed together. The result for each coin is either Heads (H) or Tails (T). The sample space diagram below shows the outcome for each coin, with two of the four outcomes already filled in.
Coin 1: H
Coin 1: T
Coin 2: H
HH
Coin 2: T
TT
(a)Complete the sample space diagram to show all four possible outcomes.(2)
(b)Write down the probability of getting two heads.(1)
(c)Find the probability of getting at least one head.(2)
(Total for Question 1 is 5 marks)
2
A fair spinner has three equal sections numbered 1, 2 and 3. A fair coin is tossed. The spinner is spun once and the coin is tossed once, and the two results are recorded together.
(a)List, as ordered pairs (spinner score, coin result), all the possible outcomes.(2)
(b)Write down the probability that the spinner lands on an odd number and the coin shows heads.(2)
(Total for Question 2 is 4 marks)
3
Spinner A has four equal sections numbered 1, 2, 3, 4. Spinner B has three equal sections numbered 1, 2, 3. Both spinners are spun together and the two scores are added. The sample space diagram below shows the sum of the two scores for some of the outcomes.
Sum
A: 1
A: 2
A: 3
A: 4
B: 1
2
3
5
B: 2
4
5
6
B: 3
4
6
7
(a)Complete the sample space diagram.(3)
(b)Write down the probability that the sum of the two scores is exactly 5.(2)
(c)Find the probability that the sum of the two scores is greater than 5.(2)
(Total for Question 3 is 7 marks)
4
Two fair six-sided dice, each numbered 1 to 6, are rolled together and the two scores are added. The sample space diagram below shows the sum of the two scores for some of the outcomes.
Sum
D1: 1
D1: 2
D1: 3
D1: 4
D1: 5
D1: 6
D2: 1
2
3
4
6
7
D2: 2
3
5
6
7
8
D2: 3
4
5
6
7
8
D2: 4
6
7
8
10
D2: 5
6
7
9
10
11
D2: 6
7
9
10
11
12
(a)Complete the sample space diagram.(3)
(b)Write down the probability that the sum of the two scores is 7.(2)
(c)Find the probability that the sum of the two scores is greater than 8.(2)
(Total for Question 4 is 7 marks)
5
Two fair six-sided dice are rolled together and the two scores are added. The sample space diagram below has already been completed, showing the sum of the two scores for every possible outcome.
Sum
D1: 1
D1: 2
D1: 3
D1: 4
D1: 5
D1: 6
D2: 1
2
3
4
5
6
7
D2: 2
3
4
5
6
7
8
D2: 3
4
5
6
7
8
9
D2: 4
5
6
7
8
9
10
D2: 5
6
7
8
9
10
11
D2: 6
7
8
9
10
11
12
(a)Use the sample space diagram to find the probability that the sum of the two scores is at least 10.(2)
(b)List, as ordered pairs (first score, second score), all the outcomes for which the sum of the two scores is exactly 11.(2)
(Total for Question 5 is 4 marks)
6
Spinner A has four equal sections numbered 1, 2, 3, 4. Spinner B has three equal sections numbered 1, 2, 3. Both spinners are spun together and the two scores are multiplied. The sample space diagram below shows the product of the two scores for some of the outcomes.
Product
A: 1
A: 2
A: 3
A: 4
B: 1
1
2
4
B: 2
4
6
8
B: 3
3
9
(a)Complete the sample space diagram.(3)
(b)Write down the probability that the product of the two scores is an even number.(2)
(c)Find the probability that the product of the two scores is greater than 6.(2)
(Total for Question 6 is 7 marks)
7
A fairground stall has a game using two fair spinners. Spinner P has five equal sections numbered 1, 2, 3, 4, 5. Spinner Q has three equal sections numbered 1, 2, 3. Both spinners are spun together and the two scores are added. A player wins if the sum of the two scores is 6 or more. As part of planning the game, the stallholder must first check that the sample space gives 15 equally likely outcomes.
Sum
P: 1
P: 2
P: 3
P: 4
P: 5
Q: 1
2
3
5
6
Q: 2
4
5
6
7
Q: 3
4
5
6
7
(a)Write down one assumption needed for the 15 outcomes in the sample space to be equally likely.(1)
(b)Complete the sample space diagram to show the sum of the two scores for every outcome.(3)
(c)Find the probability that a player wins on one go (the sum of the two scores is 6 or more).(2)
(d)The stallholder says: "About 2 in 5 games are won, so if 60 people play, I expect about 24 winners." Comment on whether this is a reasonable use of the probability from part (c).(2)
(Total for Question 7 is 8 marks)
8
The diagram shows the sample space for two fair spinners, plotted as coordinate points. Spinner A (horizontal axis) has four equal sections numbered 1 to 4. Spinner B (vertical axis) has three equal sections numbered 1 to 3. Each point marks one possible outcome (Spinner A score, Spinner B score).
(a)Use the diagram to write down the total number of possible outcomes.(1)
(b)Use the diagram to find the probability that the Spinner A score is greater than the Spinner B score.(3)
(c)Find the probability that the two scores are equal.(1)
(Total for Question 8 is 5 marks)
9
A student wants to test whether the sums of two fair dice behave as the theoretical sample space predicts. As part of this statistical enquiry, the student rolls two fair six-sided dice together 90 times and records the sum each time. The bar chart shows the results, grouped into three categories.
(a)Using a sample space diagram for two fair dice, show that the theoretical probability that the sum of the two scores is 7 is 1/6.(2)
(b)Use the bar chart to write down the number of times a sum of 7 was recorded in the 90 rolls.(1)
(c)Work out the expected number of times a sum of 7 would occur in 90 rolls, using the theoretical probability from part (a).(2)
(d)Comment on how the experimental frequency in part (b) compares with the expected frequency in part (c).(2)
(Total for Question 9 is 7 marks)
10
Spinner C has five equal sections numbered 2, 4, 6, 8, 10. Spinner D has four equal sections numbered 1, 3, 5, 7. Both spinners are spun together and the two scores are added. The sample space diagram below shows the sum of the two scores for some of the outcomes.
Sum
C: 2
C: 4
C: 6
C: 8
C: 10
D: 1
3
5
9
11
D: 3
7
9
11
13
D: 5
7
9
11
13
D: 7
9
13
15
17
(a)Complete the sample space diagram.(3)
(b)Find the probability that the sum of the two spinners is at least 13.(2)
(c)The two spinners are spun together 200 times. Work out the expected number of times the sum is at least 13.(2)
(Total for Question 10 is 7 marks)
11
Two fair four-sided dice, each numbered 1 to 4, are rolled together and the two scores are added. The sample space diagram below shows the total for every possible outcome.
Total
Die 2: 1
Die 2: 2
Die 2: 3
Die 2: 4
Die 1: 1
2
3
4
5
Die 1: 2
3
4
5
6
Die 1: 3
4
5
6
7
Die 1: 4
5
6
7
8
(a)Use the table to find the probability that the total is at most 4.(2)
(b)Use the table to find the probability that the total is at least 7.(2)
(c)Hence find the probability that the total is neither at most 4 nor at least 7 (that is, the total is 5 or 6).(2)
(Total for Question 11 is 6 marks)
12
A fair six-sided dice numbered 1 to 6 is rolled, and a fair spinner with four equal sections labelled 5, 10, 15, 20 is spun. The dice score and the spinner score are added together. Complete the sample space diagram below to show the sum for every one of the 24 outcomes.
Sum
Dice: 1
Dice: 2
Dice: 3
Dice: 4
Dice: 5
Dice: 6
Spinner: 5
Spinner: 10
Spinner: 15
Spinner: 20
(a)Complete the sample space diagram to show the sum of the dice score and the spinner score for every outcome.(4)
(b)Write down the probability that the sum is exactly 21.(2)
(c)Find the probability that the sum is greater than 20.(2)
(Total for Question 12 is 8 marks)
13
Two fair six-sided dice, each numbered 1 to 6, are rolled together. The positive difference between the two scores is recorded each time. The sample space diagram below shows the difference for some of the outcomes.
Difference
D2: 1
D2: 2
D2: 3
D2: 4
D2: 5
D2: 6
D1: 1
0
1
2
4
5
D1: 2
1
0
1
2
3
D1: 3
2
0
1
2
3
D1: 4
3
2
1
0
2
D1: 5
3
2
1
0
1
D1: 6
5
4
2
1
0
(a)Complete the sample space diagram.(3)
(b)Write down the probability that the difference between the two scores is 0.(1)
(c)Find the probability that the difference between the two scores is at least 3.(2)
(d)Two players play a game: Player A wins if the difference is 0 or 1; otherwise Player B wins. Use probabilities to determine whether this is a fair game.(3)
(Total for Question 13 is 9 marks)
14
Spinner E has three equal sections numbered 2, 4, 6. Spinner F has four equal sections numbered 1, 3, 5, 7. Both spinners are spun together and the two scores are added.
(a)List, as ordered pairs (Spinner E score, Spinner F score), all the outcomes for which the sum of the two scores is a prime number.(3)
(b)Hence write down the probability that the sum of the two spinners is a prime number.(1)
(c)The two spinners are spun together 96 times. Work out the expected number of times the sum is a prime number.(2)
(Total for Question 14 is 6 marks)
15
Spinner G has five equal sections. Two of the sections are both labelled 2; the other three sections are labelled 1, 3 and 4. Spinner H has three equal sections labelled 1, 3, 5. Both spinners are spun together and the two scores are added. A player wins if the sum of the two scores is at least 6.
Sum
Section a (1)
Section b (2)
Section c (2)
Section d (3)
Section e (4)
Spinner H: 1
2
3
4
Spinner H: 3
5
5
7
Spinner H: 5
6
7
8
9
(a)Explain why there are 15 equally likely outcomes when spinner G and spinner H are spun together, even though spinner G shows only four different numbers.(2)
(b)Complete the sample space diagram to show the sum of the two spinners for all 15 outcomes.(3)
(c)Find the probability that the sum of the two spinners is at least 6.(2)
(d)The game is played 150 times. Work out the expected number of times the sum is at least 6.(2)
(Total for Question 15 is 9 marks)
Mark scheme · S22 Sample Space Diagrams
Question 1
(a) B1 HT written in the Coin 2: H row, Coin 1: T column
(a) B1 TH written in the Coin 2: T row, Coin 1: H column
(a) Answer: HT, TH
(b) B1 1/4 oe cao
(b) Answer: 1/4
(c) M1 identifies the 3 outcomes with at least one head (HH, HT, TH), oe
(c) A1 3/4 cao
(c) Answer: 3/4
Question 2
(a) B1 at least 4 of the 6 ordered pairs listed correctly with no repeats
(c) M1 identifies the 6 outcomes with sum 6 or more, oe
(c) A1 2/5 oe cao
(c) Answer: 2/5
(d) B1 shows 60 x 2/5 = 24, matching the stallholder's claim
(d) B1 explains this is a reasonable estimate (expected value) but not a guaranteed exact outcome, since each go is random/independent, oe
(d) Answer: Reasonable estimate of 24 winners, but the actual number could differ by chance.
Question 8
(a) B1 12 cao
(a) Answer: 12
(b) B1 identifies 6 points where the Spinner A score is greater than the Spinner B score
(b) M1 6/12 oe
(b) A1 1/2 cao
(b) Answer: 1/2
(c) B1 1/4 oe cao
(c) Answer: 1/4
Question 9
(a) M1 identifies 6 outcomes with sum 7 out of 36 total outcomes, oe
(a) A1 6/36 = 1/6 shown
(a) Answer: 1/6
(b) B1 14 cao
(b) Answer: 14
(c) M1 90 x 1/6, oe
(c) A1 15 cao
(c) Answer: 15
(d) B1 notes that 14 is close to (but not exactly) the expected 15
(d) B1 explains the small difference is due to chance in a finite number of trials; the relative frequency would be expected to get closer to 1/6 with many more rolls, oe
(d) Answer: 14 is close to the expected 15; the small gap is due to chance.