12.1 Mutually exclusive and independent events — Practise

Work through these on your own. For each question, first decide whether the events can overlap, and whether they are independent.

The questions and the answers below come from a set that is checked by computer every time the book is built.

A fair six-sided die is rolled once. Find the probability of an even number or a 5.
A fair six-sided die is rolled once. Find the probability of an even number or a prime number.
A bag has 4 red counters and 6 black counters. One counter is taken at random. Find the probability that it is not red.
A fair coin is tossed and a fair six-sided die is rolled. The two results are independent. Find the probability of a head and a 6.
A card is taken at random from a standard pack of 52 cards, then replaced. A second card is taken at random. Find the probability that both cards are hearts.
A survey of 300 adults records whether each owns a car and whether each owns a bicycle. 180 own a car, 120 own a bicycle, and 72 own both. Show that owning a car and owning a bicycle are independent.
A survey of 250 people records whether each owns a cat and whether each owns a dog. 150 own a cat, 100 own a dog, and 30 own both. Show that owning a cat and owning a dog are not independent.
Two events $A$ and $B$ have $P(A) = 0.3$ and $P(B) = 0.4$ , and they are mutually exclusive. Work out $P(A) \times P(B)$ , and explain why this shows that the events are not independent.
In a class of 30 students, 18 play football, 12 play tennis, and 6 play both. A student is chosen at random. Find the probability that the student plays football or tennis.
Events $A$ and $B$ are independent with $P(A) = 0.5$ and $P(B) = 0.4$ . Find $P(A \text{ or } B)$ .
A component passes inspection with probability $0.973$ . Two components are inspected independently. Find the probability that both pass, to 4 decimal places.
A fair coin is tossed twice. Event $A$ is a head on the first toss. Event $B$ is a head on the second toss. Are $A$ and $B$ mutually exclusive? Are they independent? Explain your answers.

Answers and concise solutions

1. The probability of an even number or a 5

Answer: An even number and a 5 cannot happen together, so the events are mutually exclusive. Add the probabilities: $\dfrac{3}{6} + \dfrac{1}{6} = \dfrac{4}{6} = \dfrac{2}{3}$ .

2. The probability of an even number or a prime number

Answer: The evens are 2, 4 and 6, and the primes are 2, 3 and 5. The number 2 is in both, so subtract it once: $\dfrac{3}{6} + \dfrac{3}{6} - \dfrac{1}{6} = \dfrac{5}{6}$ . The union is 2, 3, 4, 5, 6.

3. The probability that the counter is not red

Answer: The complement rule gives $P(\text{not red}) = 1 - P(\text{red}) = 1 - \dfrac{4}{10} = \dfrac{6}{10} = \dfrac{3}{5}$ . This is also the probability of black, because every counter is red or black.

4. The probability of a head and a 6

Answer: For independent events, multiply: $P(\text{head and } 6) = P(\text{head}) \times P(6) = \dfrac{1}{2} \times \dfrac{1}{6} = \dfrac{1}{12}$ .

5. The probability that both cards are hearts

Answer: A heart is drawn with probability $\dfrac{13}{52} = \dfrac{1}{4}$ . The first card is replaced, so the draws are independent: $\dfrac{1}{4} \times \dfrac{1}{4} = \dfrac{1}{16}$ .

6. The test for independence

Answer: For independence, compare $P(\text{both})$ with $P(\text{car}) \times P(\text{bicycle})$ . Here $P(\text{both}) = \dfrac{72}{300} = \dfrac{6}{25}$ and $\dfrac{180}{300} \times \dfrac{120}{300} = \dfrac{6}{25}$ . The two agree, so the events are independent.

7. The test for independence

Answer: Here $P(\text{both}) = \dfrac{30}{250} = \dfrac{3}{25}$ , while $P(\text{cat}) \times P(\text{dog}) = \dfrac{150}{250} \times \dfrac{100}{250} = \dfrac{6}{25}$ . The two values differ, so the events are not independent. Fewer people own both than independence would predict.

8. The product of the probabilities

Answer: $P(A) \times P(B) = 0.3 \times 0.4 = 0.12 = \dfrac{3}{25}$ . Mutually exclusive events cannot happen together, so $P(A \text{ and } B) = 0$ . Since $0 \ne 0.12$ , the events are not independent. Knowing that $A$ happened makes $B$ impossible.

9. The probability that the student plays football or tennis

Answer: Use the addition rule and subtract the overlap once: $\dfrac{18 + 12 - 6}{30} = \dfrac{24}{30} = \dfrac{4}{5}$ . The 6 students who play both were counted in each total.

10. The probability of A or B

Answer: First find the overlap. Independence gives $P(A \text{ and } B) = 0.5 \times 0.4 = 0.2$ . Then $P(A \text{ or } B) = 0.5 + 0.4 - 0.2 = 0.7 = \dfrac{7}{10}$ .

11. The probability that both components pass

Answer: The passes are independent, so multiply: $0.973 \times 0.973 = 0.946729$ . To 4 decimal places this is $0.9467$ .

12. Mutually exclusive and independent

Answer: The two events are not mutually exclusive, because both can happen: the coins can show head, head. They are independent, because $P(A \text{ and } B) = \dfrac{1}{4}$ and $P(A) \times P(B) = \dfrac{1}{2} \times \dfrac{1}{2} = \dfrac{1}{4}$ . The first toss does not change the chance of a head on the second.