A bag contains 5 red counters and 3 blue counters. A counter is chosen at random. Write down the probability that the counter is red.
(1)
2
A fair six-sided die is rolled once. What is the probability of rolling an even number?
(2)
3
A spinner is split into 8 equal sectors. Two adjacent sectors are shaded. Write down the probability of landing on a shaded sector.
(2)
4
A bag contains 4 green, 2 yellow and 4 black counters. One counter is chosen at random. Calculate the probability that the counter is not yellow.
(2)
5
A coin is flipped twice. List all possible outcomes and find the probability of getting exactly one head.
(3)
6
Two dice are rolled. What is the probability that the total is 7?
(2)
7
A bag contains 3 red, 3 white and 4 blue counters. Two counters are drawn at random without replacement. Calculate the probability that the first is red and the second is blue.
(3)
8
A box contains 6 white and 2 black balls. A ball is drawn, its colour noted, then it is replaced. This experiment is repeated twice. Calculate the probability of drawing black then white in that order.
(2)
9
A card is drawn at random from a set of 12 cards numbered 1 to 12. What is the probability the number is a multiple of 3?
(2)
10
A spinner has three equal sectors labelled A, B and C. The spinner is spun twice. Find the probability of getting A at least once.
(3)
11
A bag contains 7 beads: 3 gold, 2 silver and 2 copper. One bead is picked at random. Give the probability that the bead is metal-coloured (silver or copper).
(2)
12
A school has 40 students in a class. 25 students take music, 18 take art and 10 take both music and art. A student is chosen at random. Calculate the probability the student takes music or art.
(3)
13
A fair 10-sided spinner is numbered 0 to 9. Two independent spins are made. What is the probability that the sum of the two numbers is 10?
(2)
14
A jar contains 5 orange sweets and 7 lemon sweets. Two sweets are eaten at random without replacement. Find the probability both sweets are the same flavour.
(3)
15
A box contains slips labelled A, B, C, D, E. One slip is chosen at random and then replaced. This is repeated three times. Calculate the probability that letter A appears exactly once in the three draws.
(3)
16
A fair coin is flipped until heads appears. What is the probability that heads appears for the first time on the third flip?
(4)
17
A fair six-sided die is rolled. Given the result is an even number, what is the probability that it is a 6?
(3)
18
Stretch question. A game uses two fair coins and one six-sided die. A player wins the prize if they get at least two heads from the two coins and roll a number greater than 4 on the die. Find the probability of winning.
(4)
Mark scheme · KS3.M-P8 Mixed Probability Problem Set
Question 1
B1 5/8 cao
Answer: 5/8
Question 2
M1 correct identification of 3 favourable outcomes (2,4,6) out of 6 or equivalent method
A1 1/2 or 3/6 cao
Answer: 1/2
Question 3
M1 recognition that probability = shaded sectors / total sectors
A1 2/8 = 1/4 cao
Answer: 1/4
Question 4
M1 correct total count 10 and recognition of non-yellow count 8
A1 8/10 = 4/5 cao
Answer: 4/5
Question 5
M1 correct list of outcomes: HH, HT, TH, TT or equivalent
M1 identification of favourable outcomes HT and TH
A1 2/4 = 1/2 cao
Answer: 1/2
Question 6
M1 recognition of 6 favourable outcomes out of 36
A1 6/36 = 1/6 cao
Answer: 1/6
Question 7
M1 probability first red = 3/10 and then blue = 4/9 or equivalent
M1 product method used: (3/10)*(4/9)
A1 12/90 = 2/15 cao
Answer: 2/15
Question 8
M1 recognition that replacement keeps probabilities the same: black 2/8, white 6/8
A1 (2/8)*(6/8) = 12/64 = 3/16 cao
Answer: 3/16
Question 9
M1 identification of multiples of 3: 3,6,9,12 gives 4 favourable out of 12
A1 4/12 = 1/3 cao
Answer: 1/3
Question 10
M1 recognition that easier to use complement: probability no A in two spins = (2/3)*(2/3)
M1 calculate (2/3)2 = 4/9
A1 1 - 4/9 = 5/9 cao
Answer: 5/9
Question 11
M1 recognition silver or copper count = 4 out of 7
A1 4/7 cao
Answer: 4/7
Question 12
M1 use of inclusion-exclusion: 25 + 18 - 10 = 33 students take music or art
M1 recognition total 40 and forming probability 33/40
A1 33/40 cao
Answer: 33/40
Question 13
M1 count pairs (0,10) invalid so valid ordered pairs are (1,9),(2,8),(3,7),(4,6),(5,5),(6,4),(7,3),(8,2),(9,1) = 9 pairs out of 100
A1 9/100 cao
Answer: 9/100
Question 14
M1 probability both orange = (5/12)*(4/11) and both lemon = (7/12)*(6/11) recognised
M1 sum of probabilities computed: (20/132)+(42/132) = 62/132
A1 62/132 = 31/66 cao
Answer: 31/66
Question 15
M1 probability of A on one draw = 1/5 and not A = 4/5 used and recognition of binomial count 3 positions
M1 compute 3*(1/5)*(4/5)2
A1 3*(1/5)*(16/25) = 48/125 cao
Answer: 48/125
Question 16
M1 recognition that first two flips must be tails with probability (1/2)*(1/2)
M1 recognition third flip must be heads with probability 1/2
M1 product method used: (1/2)3
A1 1/8 cao
Answer: 1/8
Question 17
M1 recognition conditional sample space of even numbers {2,4,6} size 3
M1 favourable outcome 6 is one of these
A1 1/3 cao
Answer: 1/3
Question 18
M1 calculate probability of at least two heads from two coins: outcomes HH, HT, TH, TT so P(at least two heads)=P(HH)=1/4
M1 calculate probability die > 4: favourable 5 or 6 gives 2/6 = 1/3
M1 recognition of independence and multiplication of probabilities