A fair six-sided dice is rolled once. State whether the events 'rolling a 2' and 'rolling a 5' are mutually exclusive.
(Total for Question 1 is 1 mark)
2
A card is drawn from a standard 52-card pack. State whether the events 'drawing a King' and 'drawing a Heart' are mutually exclusive.
(Total for Question 2 is 1 mark)
3
Events X and Y are mutually exclusive. P(X) = 0.4 and P(Y) = 0.35. Write down the value of P(X or Y).
(Total for Question 3 is 1 mark)
4
State the mathematical term for two or more events that, between them, cover every possible outcome.
(Total for Question 4 is 1 mark)
5
A bag has red, blue and green counters only. P(red) = 0.4, P(blue) = 0.25 and P(green) = 0.35. Explain how you know that these three colours are all of the colours in the bag.
(Total for Question 5 is 2 marks)
6
A spinner has sections A, B and C only. P(A) = 0.3 and P(B) = 0.45. Work out P(C).
(Total for Question 6 is 2 marks)
7
Events P and Q are mutually exclusive. P(P) = 0.18 and P(Q) = 0.37. Work out P(P or Q).
(Total for Question 7 is 2 marks)
8
A biased four-sided spinner has sections labelled 1, 2, 3 and 4. P(1) = 0.15, P(2) = 0.2 and P(3) = 0.4. Work out P(4).
(Total for Question 8 is 2 marks)
9
A fair 12-sided dice, numbered 1 to 12, is rolled once. No number from 1 to 12 is a multiple of both 4 and 5. Work out the probability of rolling a multiple of 4 or a multiple of 5.
(Total for Question 9 is 2 marks)
10
In a class of 32 students, every student studies exactly one of French, Spanish or German. 14 study French and 9 study Spanish. Work out the probability that a student picked at random studies German.
(Total for Question 10 is 2 marks)
11
A game show spinner has three mutually exclusive and exhaustive outcomes: Win, Draw and Lose. P(Win) = 0.2 and P(Lose) = 0.55. Work out P(Draw).
(Total for Question 11 is 2 marks)
12
A survey found that P(a person owns a cat) = 0.3 and P(a person owns a dog) = 0.45, and some people in the survey own both a cat and a dog. Write down why 'owns a cat' and 'owns a dog' are not mutually exclusive events in this survey.
(Total for Question 12 is 1 mark)
13
A fair spinner has sections numbered 1 to 6. Write down whether the events 'landing on an odd number' and 'landing on an even number' are mutually exclusive.
(Total for Question 13 is 1 mark)
14
A bag contains only black and white counters, 40 counters in total. The probability of picking a black counter at random is 0.6. Work out the number of white counters in the bag.
(Total for Question 14 is 2 marks)
15
A four-sided spinner has sections P, Q, R and S, with probabilities in the ratio 1:2:3:4 respectively. Work out P(S).
(Total for Question 15 is 3 marks)
16
In a survey of 45 people, event A is 'drinks tea' and event B is 'drinks coffee'. No person in the survey drinks both. 27 people drink tea and 11 drink coffee. Work out the probability that a randomly chosen person drinks neither tea nor coffee, and give this probability as a percentage to 1 decimal place.
(Total for Question 16 is 3 marks)
17
A biased six-sided dice has probabilities for scores 1, 2, 3, 4, 5, 6 given by 2x, 2x, x, x, 3x, x respectively. Use the fact that the probabilities sum to 1 to find x, then work out P(rolling a 5).
(Total for Question 17 is 3 marks)
18
A charity shop's payments are either Card or Cash, which are mutually exclusive and exhaustive. In one week, 160 items were sold, and the probability that an item was paid for by card is 0.625. Work out the number of items paid for by cash.
(Total for Question 18 is 3 marks)
19
A class has 30 students. Event M is 'studies Music' and event D is 'studies Drama'. 12 students study Music only, 7 study Drama only, and 5 study both Music and Drama. Work out the probability that a randomly chosen student studies neither Music nor Drama.
(Total for Question 19 is 3 marks)
20
A gym reports that 65% of members use the pool and 50% use the gym floor. The manager claims: "Since 65%+50%=115%, this proves at least 15% of members use both facilities." Explain whether the manager's reasoning is valid, and show the minimum percentage of members who must use both.
(Total for Question 20 is 3 marks)
Mark scheme · S20D Mutually Exclusive Events: Fluency and Exam Drill
Question 1
B1 mutually exclusive - a single roll cannot be both 2 and 5, oe
Answer: Mutually exclusive
Question 2
B1 not mutually exclusive - the King of Hearts is both a King and a Heart, oe
Answer: Not mutually exclusive
Question 3
B1 0.75 cao
Answer: 0.75
Question 4
B1 exhaustive cao
Answer: Exhaustive
Question 5
M1 0.4 + 0.25 + 0.35 (= 1), oe
A1 since the probabilities sum to 1, these outcomes are exhaustive, so no other colour is possible, oe
Answer: 0.4+0.25+0.35 = 1, so red, blue and green must be the only possible colours (they are exhaustive).
Question 6
M1 1 - 0.3 - 0.45, oe
A1 0.25 cao
Answer: 0.25
Question 7
M1 0.18 + 0.37, oe
A1 0.55 cao
Answer: 0.55
Question 8
M1 1 - 0.15 - 0.2 - 0.4, oe
A1 0.25 cao
Answer: 0.25
Question 9
M1 identifies 4, 8, 12 (multiples of 4) and 5, 10 (multiples of 5), 5 outcomes with no overlap
A1 5/12 oe cao
Answer: 5/12
Question 10
M1 32 - 14 - 9, oe
A1 9/32 oe cao
Answer: 9/32
Question 11
M1 1 - 0.2 - 0.55, oe
A1 0.25 cao
Answer: 0.25
Question 12
B1 some people own both a cat and a dog, so the two events can happen at the same time, oe
Answer: Some people own both a cat and a dog, so the two events can occur together.
Question 13
B1 mutually exclusive cao
Answer: Mutually exclusive
Question 14
M1 (1 - 0.6) x 40, oe
A1 16 cao
Answer: 16
Question 15
M1 total parts = 1+2+3+4 (= 10), oe
M1 S's share = 4/10, oe
A1 2/5 oe cao
Answer: 2/5
Question 16
M1 45 - 27 - 11 (= 7), oe
A1 7/45 oe cao
B1 15.6% (1 d.p.) cao ft
Answer: 7/45, which is 15.6% (1 d.p.)
Question 17
M1 2x+2x+x+x+3x+x = 1, i.e. 10x = 1, oe
M1 x = 0.1
A1 0.3 cao
Answer: 0.3
Question 18
M1 0.625 x 160 (= 100), oe
M1 160 - 100, oe
A1 60 cao
Answer: 60
Question 19
M1 12 + 7 + 5 (= 24), oe
M1 30 - 24, oe
A1 1/5 oe cao
Answer: 1/5
Question 20
M1 recognises that the probability of using at least one facility cannot exceed 100%
A1 65+50-100 = 15 shown, oe
B1 concludes the manager's reasoning is valid: at least 15% of members must use both facilities
Answer: The manager's reasoning is valid. Since P(pool or gym floor) cannot exceed 100%, P(both) ≥ 65+50-100 = 15%, so at least 15% of members must use both.