**Question 1.**

A survey obtained the following results: 42 people had been to the Moon, 57 had been to Mars, 52 had been to Venus, 17 had been to the Moon and Mars, 12 had been to the Moon and Venus, 7 had been to Mars and Venus, 4 had been to all three, and 18 had been to none of the three.

(a) How many people were interviewed?

(b) How many people had only been to Mars?

(c) How many people had only been to one of the three places? Hint: Use a Venn diagram.

**Question 2.**

(a) A pair of identical fair dice is rolled. What is the probability that the sum of the numbers is more than 6?

(b) A pair of identical fair dice is rolled. What is the probability that one of the numbers is more than 4?

(c) A pair of identical fair dice is rolled. What is the probability that the sum of the numbers is more than 6, if it is known that one of the numbers is a 4?

(d) A pair of fair dice, one red and one green, is rolled. What is the probability that the sum of the numbers is more than 6, if it is known that the red one shows a 4?

**Question 3.**

Assume that the probability of a boy being born is the same as the probability of a girl being born.

(a) Find the probability that for family with four children, the two oldest children are girls.

(b) Find the probability that for family with four children, exactly two of the children are boys.

(c) Find the probability that for family with four children, at least two of the children are boys.

**Question 4.**

A survey shows that 80% of a population has been vaccinated against the flu, but 5% of the vaccinated population gets the flu anyway. In total 10% of the population gets the flu. Let V be the event that a randomly selected person in the population has been vaccinated and Fthe event that a randomly selected person in the population gets the flu.

(a) Present the information using either a Venn diagram or a table or both.

(b) Estimate the following probabilities and describe the probabilities in words.

• P(F')

• F(F∩V)

• P(F∩V')

• P(F|V)

• P(V|F)

(c) Does this survey provide evidence that the events F and V are not independent? Give a reason for your answer.

(d) Which conditional probabilities will you calculate to explain whether vaccination should be encouraged?

**Question 5.**

Three horses A, B and C are going to race against each other. The probability that A wins is 1/5. The probability that B wins is 1/2

(a) What is the probability that C wins?

(b) What are the odds that C wins?

(c) Are the events A wins and B wins mutually exclusive? Give a reason for your answer.

(d) Are the events A wins and B wins independent? Give a reason for your answer.

**Question 6.**

HTML colours are represented by 6-digit hexadecimal codes.

Each digit can take on 1 of 16 values: 0, 1, 2, . . . , A, B, C, D, E, F.

(a) How many different colours can be represented?

(b) There are three types of pure colours: pure red (represented by xy0000); pure green (represented by 00xy00); pure blue (represented by 0000.4. How many pure colours are possible?

(c) Grayscale shades are represented by codes xyxyxy consisting of a repeated pair of digits. How many grayscale shades are possible?

(d) Some monitors could only display colours with codes consisting of three pairs of repeating digits: xxyyzz. How many colours could these monitors display?

**Question 7.**

There are 65 handmade music boxes, four of which are made by an assistant and 61 are made by technicians. Ten randomly selected music boxes are tested.

(a) In how many ways can ten music boxes be selected from the 65 music boxes? Give the answer in factorial notation.

(b) What is the probability that at least one music box made by the assistant is selected?

**Question 8.**

Three marbles are removed from a box that contains 4 red and 8 green marbles.

(a) What is the probability that all three are red?

(b) What is the probability that all three are of the same colour?

(c) What is the probability that at least 1 marble is red?

**Question 9.**

Assume that E and F are two mutually exclusive events. Will the events E' and P be mutually exclusive? Give a reason for your answer.

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