O Level Mathematics D (4024)
Probability Banker
36 questions · about 4% of the marks we indexed · examined in 9 of 9 sessions.
Practice questions
Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not
Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.
Q1Combined events · Core · Paper 1 style · no calculator · 6 marksA fair spinner is numbered 1, 2, 3, 4. A fair six-sided die is numbered 1 to 6. The spinner is spun and the die is rolled. The score is the spinner number multiplied by the die number.
- (a) Find the probability that the score is 12. [2]
- (b) Find the probability that the score is odd. [2]
- (c) Find the probability that the score is greater than 15. [2]
Show answer
- (a) 1/8
Pairs (2, 6), (3, 4), (4, 3) M1; 3/24 = 1/8 A1
- (b) 1/4
Both odd: 2/4 × 3/6 M1; 1/4 A1
- (c) 1/6
16, 18, 20, 24: 4 outcomes M1; 4/24 = 1/6 A1
Two fair four-sided spinners are each numbered 1, 2, 3 and 4. Both are spun and the two numbers are added.
- (a) Find the probability that the total is 5. [2]
- (b) Find the probability that the total is 7 or more. [1]
- (c) Find the probability that both spinners show the same number. [1]
Show answer
- (a) 1/4
16 outcomes; (1,4),(2,3),(3,2),(4,1) M1 A1
- (b) 3/16
(3,4),(4,3),(4,4) B1
- (c) 1/4
B1
Usman takes a bus and then a train to work. The probability that the bus is late is 0.2. The probability that the train is late is 0.15. These events are independent.
- (a) Find the probability that both the bus and the train are late. [1]
- (b) Find the probability that neither is late. [2]
- (c) Find the probability that exactly one of them is late. [2]
Show answer
- (a) 0.03
0.2 × 0.15 B1
- (b) 0.68
0.8 × 0.85 M1 A1
- (c) 0.29
0.2 × 0.85 + 0.8 × 0.15 M1 A1
A box contains 3 green balls and 7 yellow balls. A ball is taken at random, its colour noted, and it is replaced. A second ball is then taken at random.
- (a) Find the probability that both balls are green. [2]
- (b) Find the probability that at least one ball is yellow. [2]
Show answer
- (a) 9/100
3/10 × 3/10 M1 A1
- (b) 91/100
1 − 9/100 M1 A1
A fair coin is thrown and a biased spinner is spun. The spinner lands on red with probability 0.35, blue with probability 0.4 and green with probability 0.25.
- (a) Find the probability of getting a head and green. [1]
- (b) Find the probability of getting a tail and a colour that is not red. [2]
Show answer
- (a) 0.125
B1
- (b) 0.325
0.5 × 0.65 M1 A1
A bag contains 5 red counters and 3 blue counters. Two counters are taken at random without replacement.
- (a) Find the probability that both counters are red. [2]
- (b) Find the probability that the two counters are different colours. [2]
- (c) Find the probability that at least one counter is blue. [1]
Show answer
- (a) 5/14
5/8 × 4/7 M1; 20/56 = 5/14 A1
- (b) 15/28
5/8 × 3/7 + 3/8 × 5/7 M1; 30/56 = 15/28 A1
- (c) 9/14
1 − 5/14 B1
The probability that Ahmed passes his driving test at the first attempt is 0.6. If he fails, the probability that he passes at the second attempt is 0.7.
- (a) Find the probability that Ahmed passes on his second attempt. [2]
- (b) Find the probability that he passes within two attempts. [2]
- (c) If he fails twice, he stops. Find the probability that he stops without passing. [1]
Show answer
- (a) 0.28
0.4 × 0.7 M1 A1
- (b) 0.88
0.6 + 0.28 M1 A1
- (c) 0.12
0.4 × 0.3 B1
A box contains 10 chocolates: 4 dark and 6 milk. Sana takes three chocolates at random without replacement.
- (a) Find the probability that all three chocolates are milk. [2]
- (b) Find the probability that at least one chocolate is dark. [1]
- (c) Find the probability that exactly one chocolate is dark. [3]
Show answer
- (a) 1/6
6/10 × 5/9 × 4/8 M1 A1
- (b) 5/6
1 − 1/6 B1
- (c) 1/2
4/10 × 6/9 × 5/8 M1; × 3 orders M1; A1
In a class of 32 students, 18 play cricket, 14 play football and 6 play both.
- (a) Find the number of students who play neither sport. [1]
- (b) A student is chosen at random. Find the probability that the student plays exactly one sport. [2]
- (c) A student who plays football is chosen at random. Find the probability that this student also plays cricket. [2]
Show answer
- (a) 6
32 − 26 B1
- (b) 5/8
12 + 8 = 20 M1; 20/32 A1
- (c) 3/7
6 out of 14 M1 A1
Two fair six-sided dice are rolled. The score is the larger number minus the smaller number (0 if they are equal).
- (a) Find the probability that the difference is 0. [1]
- (b) Find the probability that the difference is 3 or more. [2]
- (c) Find the most likely difference. [1]
Show answer
- (a) 1/6
B1
- (b) 1/3
Difference 3: 6, 4: 4, 5: 2 ways M1; 12/36 A1
- (c) 1
10 ways out of 36 B1
200 people were asked whether they prefer tea or coffee.Under 30: tea 46, coffee 3430 and over: tea 78, coffee 42
- (a) One person is chosen at random. Find the probability that the person prefers coffee. [1]
- (b) A person who prefers coffee is chosen at random. Find the probability that this person is under 30. [2]
- (c) Two different people are chosen at random. Find the probability that both prefer tea. Give your answer correct to 3 significant figures. [2]
Show answer
- (a) 19/50
76/200 B1
- (b) 17/38
34 out of 76 M1 A1
- (c) 0.383
124/200 × 123/199 M1 A1
On any day, the probability that it is sunny is 0.65. When it is sunny, the probability that a cricket team wins its match is 0.7. When it is not sunny, the probability is 0.45. Different days are independent.
- (a) Find the probability that the team wins a match. [3]
- (b) The team plays two matches on different days. Find the probability that it wins both, correct to 3 significant figures. [2]
Show answer
- (a) 0.6125
0.65 × 0.7 M1; + 0.35 × 0.45 M1; 0.6125 A1
- (b) 0.375
0.6125² M1 A1
A drawer contains 6 black socks and 4 white socks. Two socks are taken at random without replacement.
- (a) Find the probability that both socks are black. [1]
- (b) Find the probability that the two socks are the same colour. [2]
Show answer
- (a) 1/3
6/10 × 5/9 B1
- (b) 7/15
30/90 + 12/90 M1 A1
The probability that it rains on a school day is 0.3. If it rains, the probability that Amna is late is 0.4. If it does not rain, the probability that she is late is 0.1. Days are independent.
- (a) Find the probability that Amna is late on a given day. [2]
- (b) Find the probability that she is late on exactly one of two days. [2]
- (c) Find the probability that she is not late on either of the two days. [1]
Show answer
- (a) 0.19
0.3 × 0.4 + 0.7 × 0.1 M1; 0.19 A1
- (b) 0.3078
2 × 0.19 × 0.81 M1; 0.3078 A1
- (c) 0.6561
0.81² B1
A bag contains n counters. 4 of them are red. Two counters are taken at random without replacement. The probability that both are red is 1/11.
- (a) Show that n² − n − 132 = 0. [3]
- (b) Solve the equation to find n. [2]
Show answer
- (a) 4/n × 3/(n − 1) = 1/11 gives n² − n = 132
4/n × 3/(n − 1) M1; = 1/11 M1; 12 × 11 = n(n − 1) leading to n² − n − 132 = 0 A1
- (b) n = 12
(n − 12)(n + 11) = 0 M1; n = 12 (n > 0) A1
Three archers each shoot one arrow at a target. The probabilities that they hit the target are 0.7, 0.6 and 0.8, independently.
- (a) Find the probability that all three hit the target. [1]
- (b) Find the probability that exactly two of them hit the target. [3]
- (c) Find the probability that at least one of them hits the target. [2]
Show answer
- (a) 0.336
B1
- (b) 0.452
Three cases M1; each product correct M1; 0.084 + 0.224 + 0.144 A1
- (c) 0.976
1 − 0.3 × 0.4 × 0.2 M1 A1
Sara and Bilal take turns to roll a fair six-sided die, with Sara going first. The first person to roll a 6 wins.
- (a) Find the probability that Sara wins on her first turn or her second turn. [3]
- (b) Find the probability that nobody has won after each player has had two turns. [2]
Show answer
- (a) 61/216
1/6 M1; (5/6)(5/6)(1/6) = 25/216 M1; 36/216 + 25/216 A1
- (b) 625/1296
(5/6)⁴ M1 A1
A factory makes bolts on two machines. Machine A makes 60% of the bolts and 3% of its bolts are faulty. Machine B makes the rest and 5% of its bolts are faulty.
- (a) Find the probability that a bolt chosen at random is faulty. [2]
- (b) A bolt is chosen at random and is faulty. Find the probability that it was made by machine B. Give your answer correct to 3 significant figures. [2]
Show answer
- (a) 0.038
0.6 × 0.03 + 0.4 × 0.05 M1 A1
- (b) 0.526
0.02 ÷ 0.038 M1 A1
A spinner has four colours. It is spun 200 times.Red: 46, Blue: 58, Green: 64, Yellow: 32
- (a) Find the relative frequency of green. [1]
- (b) The spinner is spun 500 more times. Estimate the number of times it lands on yellow. [2]
- (c) Is the spinner fair? Give a reason. [1]
Show answer
- (a) 0.32
64/200 B1
- (b) 80
32/200 = 0.16 M1; 0.16 × 500 = 80 A1
- (c) Probably not: a fair spinner would give about 50 of each colour; yellow 32 and green 64 are far from 50.
B1
The probability that a seed of a certain type germinates is 0.85. A farmer near Sahiwal plants 400 of these seeds.
- (a) How many seeds are expected to germinate? [1]
- (b) How many seeds are expected not to germinate? [1]
Show answer
- (a) 340
0.85 × 400 B1
- (b) 60
B1
A biased six-sided die has these probabilities.Score 1: 0.15, 2: 0.2, 3: x, 4: 0.1, 5: 0.25, 6: 2x
- (a) Find x. [2]
- (b) The die is rolled 150 times. Find the expected number of sixes. [2]
- (c) Find the probability of rolling an even number. [1]
Show answer
- (a) x = 0.1
0.15 + 0.2 + x + 0.1 + 0.25 + 2x = 1 M1; 3x = 0.3, x = 0.1 A1
- (b) 30
P(6) = 0.2 M1; 0.2 × 150 = 30 A1
- (c) 0.5
0.2 + 0.1 + 0.2 B1
In a traffic survey at a junction in Multan, 250 vehicles were counted.Cars: 118, Motorbikes: 87, Rickshaws: 30, Buses: 15
- (a) Find the relative frequency of a motorbike. [1]
- (b) On another day, 1800 vehicles pass the junction. Estimate the number of motorbikes. [2]
- (c) Estimate the probability that the next vehicle is not a car. [1]
Show answer
- (a) 87/250 (= 0.348)
87 ÷ 250 B1
- (b) 626.4, so about 626
0.348 × 1800 M1 A1
- (c) 66/125 (= 0.528)
B1
Bilal throws a drawing pin. In the first 40 throws it lands point up 14 times. In the next 160 throws it lands point up 66 times.
- (a) Find the relative frequency of 'point up' after the first 40 throws and after all 200 throws. [2]
- (b) Which value is the better estimate of the probability? Give a reason. [1]
- (c) The pin is thrown 750 times. Estimate how many times it lands point up. [1]
- (d) A bag contains 30 counters, some red. A counter is taken and replaced many times; the relative frequency of red settles at 0.4. Estimate the number of red counters. [2]
Show answer
- (a) 0.35 and 0.4
14/40 B1; (14 + 66)/200 B1
- (b) 0.4, because it is based on more throws.
B1
- (c) 300
0.4 × 750 B1
- (d) 12
0.4 × 30 M1; 12 A1
Hira flips a coin 50 times and gets 31 heads. Zain flips the same coin 300 times and gets 156 heads.
- (a) Find each person's relative frequency of heads. [2]
- (b) Whose result gives the more reliable estimate? Give a reason. [1]
- (c) Combine their results to estimate the probability of a head, correct to 3 significant figures. [2]
- (d) Use your answer to part (c) to estimate the number of heads in 1000 flips. [1]
Show answer
- (a) Hira 31/50 (0.62), Zain 13/25 (0.52)
B1 B1
- (b) Zain's, because he did more trials.
B1
- (c) 0.534
187 ÷ 350 M1 A1
- (d) 534
B1
A jar contains 4 red sweets, 7 blue sweets and 9 green sweets. One sweet is taken at random.
- (a) Find the probability that the sweet is blue. [1]
- (b) Find the probability that the sweet is not green. [1]
- (c) Find the probability that the sweet is yellow. [1]
Show answer
- (a) 7/20
B1
- (b) 11/20
B1
- (c) 0
B1
Each letter of the word ISLAMABAD is written on a separate card. One card is chosen at random.
- (a) Find the probability that the card shows A. [1]
- (b) Find the probability that the card does not show a vowel. [1]
- (c) Find the probability that the card shows Z. [1]
Show answer
- (a) 1/3
3 A's out of 9 cards B1
- (b) 5/9
Vowels I, A, A, A; 5 consonants B1
- (c) 0
No Z: impossible B1
A bag contains only red, white and black counters. A counter is taken at random. P(red) = 0.25 and P(white) = 0.4.
- (a) Find the probability that the counter is black. [1]
- (b) There are 14 black counters. Find the total number of counters and the number of red counters. [2]
- (c) Some more white counters are added so that the probability of white becomes 1/2. How many are added? [2]
Show answer
- (a) 0.35
1 − 0.25 − 0.4 B1
- (b) 40 counters; 10 red
14 ÷ 0.35 = 40 M1; 0.25 × 40 = 10 A1
- (c) 8
(16 + n)/(40 + n) = 1/2 M1; n = 8 A1
A whole number from 10 to 50 inclusive is chosen at random.
- (a) Find the probability that the number is a multiple of 7. [2]
- (b) Find the probability that the number is prime. [2]
- (c) Find the probability that the number is a square number. [1]
Show answer
- (a) 6/41
41 numbers B1; 14, 21, 28, 35, 42, 49 B1
- (b) 11/41
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 M1 A1
- (c) 4/41
16, 25, 36, 49 B1
A box contains x blue pens and (x + 5) red pens only. A pen is taken at random.
- (a) The probability of taking a blue pen is 2/5. Find x. [3]
- (b) Three blue pens are removed from the box. A pen is then taken at random. Find the probability that it is red. [2]
Show answer
- (a) x = 10
x/(2x + 5) = 2/5 M1; 5x = 4x + 10 M1; x = 10 A1
- (b) 15/22
15 red out of 22 pens M1 A1
A bag contains red, blue and white marbles only. The probability of taking a red marble is 0.15. There are 9 red marbles, and there are twice as many blue marbles as red marbles.
- (a) Find the total number of marbles and the probability of taking a white marble. [3]
- (b) Some white marbles are removed. The probability of taking a white marble is now ½. How many white marbles were removed? [2]
Show answer
- (a) 60 marbles; P(white) = 11/20
9 ÷ 0.15 = 60 M1; blue 18, white 60 − 27 = 33 M1; 33/60 = 11/20 A1
- (b) 6
(33 − k)/(60 − k) = ½ M1; k = 6 A1
