Exercises: Sample spaces, counting and relative frequency

A fair six-sided die is rolled once.

1. a) List the sample space.
2. b) Find the probability of rolling a number greater than 4.
3. c) Find the probability of rolling a prime number.
4. d) Find the probability of rolling a 7, and state what this tells you about the event.

A bag contains 5 red, 3 blue and 2 green counters. One counter is picked at random.

1. a) Find the probability that it is red.
2. b) Find the probability that it is not blue.
3. c) Find the probability that it is red or green.

A coin is tossed 200 times and lands on heads 94 times.

1. a) Write the relative frequency of heads as a decimal.
2. b) Write the relative frequency of tails as a decimal.
3. c) How many heads would you expect in 1,000 tosses of a fair coin?

Two fair dice are rolled and their scores are added.

1. a) Draw a two-way table showing all possible outcomes.
2. b) Find the probability that the sum is 8.
3. c) Find the probability that the sum is at least 10.
4. d) Find the probability that the sum is a multiple of 5.

A café in Cork offers 4 types of sandwich, 3 types of soup and 5 types of drink. A lunch deal consists of one sandwich, one soup and one drink.

1. a) How many different lunch deals are possible?
2. b) Aoife does not want soup. How many different sandwich-and-drink combinations can she choose?
3. c) Two of the drinks are hot. How many lunch deals include a hot drink?

The probability that Seán's bus is late on any given morning is $0.15$. The probability that it is early is $0.1$.

1. a) Find the probability that the bus is on time.
2. b) Draw a tree diagram for two consecutive mornings, assuming the bus is either late or not late and mornings are independent.
3. c) Find the probability that the bus is late on exactly one of the two mornings.

A spinner has three equal sectors labelled 1, 2 and 3. It is spun twice, and a coin is tossed once.

1. a) How many outcomes are in the sample space?
2. b) Find the probability that the two spins give the same number and the coin shows heads.
3. c) Find the probability that the sum of the two spins is 4 or more, and the coin shows tails.

A bag contains 4 red and 6 white discs. Two discs are drawn at random one after the other, without replacement.

1. a) Draw a tree diagram showing all outcomes and their probabilities.
2. b) Find the probability that the two discs are the same colour.
3. c) Find the probability that at least one disc is red.
4. d) Explain how the answers would change if the first disc were replaced before the second draw.

A car registration plate is to be made from 2 different letters chosen from $\{A, B, C, D, E\}$ followed by a 3-digit number using digits 1 to 9, where digits may repeat.

1. a) How many different plates are possible?
2. b) How many plates have an even last digit?
3. c) A plate is chosen at random. Find the probability that it contains the letter $A$.