1) For each of the following, decide whether the probability for the event is theoretical or experimental: (a) A member of your family is picked at random We want to find: p(they are female) (b) A fox is thought to be attacking the hens on a farm. We want p(fox weighs more than 20 kg) (c) A card is drawn from a pack (52 cards). We want: p(the card is an ace) (d) A buttered toast falls out of my hand. We want: p(it lands buttered side down!) Find theoretical probabilities. Describe how you would determine experimental probabilities.

2) A square board of 30 cm is marked with triangle height 25 cm (as shown). A coin is thrown onto the board; its centre is equally likely to fall anywhere on the board. Find the probability that the centre of the coin falls in the shaded region.

3) A drawer contains grey and brown socks. If a sock is picked at random the probability that it is grey is . What is the probability of picking a brown sock?

4) A box contains yellow, red and black counters. If the probability of getting a yellow counter is and of getting a red counter is , find the probability of getting a black counter.

5) The probabilities of three hockey teams and winning a tournament are and respectively. Find the probability that: (a) either and wins, (b) any one of these teams wins, (c) none of these teams wins.

6) A coin is biased so the probability of getting a tail is twice that of getting a head. What is p(head)?

7) The probability of drawing an ace from a hand of 12 cars is . How many aces are in the hand?

8) In a car park there is a probability of that a car picked at random is British. There are 144 cars in the car park. How many of them are not British?

9) From a handful of red cards and black card the probability of drawing a red card at random is . There are 24 red cards. How many black cards are there?

10) On a shelf are French and German text books. The probability is that a book picked at random is French. There are 24 German books. How many French books are there?

11) A disc is tossed on to a square board of side 1 m marked, as shown in the diagram, with a circle radius 0.4 m. The centre of the disc is equally likely to fall at any point on the board. Find the probability that the centre of the disc will fall on: (a) the un-shaded part, (b) the shaded part. (Give answers to 3 s.f.)

12) As shown in the Venn diagram, all but two of the pupil at a school chose to study one or more of the subjects; Physics, Chemistry and Biology. If a random pupil is picked, what is the probability: (a) the pupil took chemistry, (b) the pupil took chemistry but not physics or biology. (c) If a physics student is picked at random, what is the probability they took both biology and chemistry as well?

13) A counter is thrown onto a board which is marked in the diagram. Its centre is equally likely to fall at any point on the board. Find the probability that it lands on the shaded area.

14) A bag contains 24 white and some black counters. How many are black, if p(black) = .

15) A card is drawn from a pack of 52. Find the probability that it is neither a king nor black.

1) (a) A and B are mutually exclusive events and p(A) = 0.3 and p(B) = 0.4. Find p(A or B). (b) C and D are independent events and p(C) = 0.6 and p(D) = 0.4. Find p(C and D).

2) For events A and B: p(A) = 0.5 and p(B) = 0.6. Given that p(A or B) = 0.9, decide if A and B are mutually exclusive events, or not (giving a reason).

3) A fair die is rolled. List all pairs of the following that are mutually exclusive: A: Result is EVEN B: Result is ODD C: Result is PRIME D: Result is a ‘6’

4) A card is drawn from a pack of 52. Which pairs of the following events are mutually exclusive: A: Card is RED B: Card is a SPADE C: Card is an ACE D: Card is the Jack of ♥

5) For events A and B: p(A) = 0.3 and p(B) = 0.2. Given that p(A and B) = 0.08, decide if A and B are dependent events, or independent events (giving a reason).

6) A random digit is picked from: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Find probabilities for each of these events: A: Digit is ODD B: Digit is PRIME C: Digit is over 3 (b) Which pairs of events are independent?

7) A card is drawn from a pack of 52. List all pairs of the following that are independent A: Card is RED B: Card is a SPADE C: Card is an ACE

8) For each of these experiments below; (a) list all of the possible outcomes for the experiment, showing that the outcomes within the experiment are equally likely AND (b) calculate the probability for the outcome required:

i) Experiment #1: Three children are picked at random from a school register. Outcome required: They are all boys.

ii) Experiment #2: Four coins are spun. Outcome required: The result is more heads than tails.

iii) Experiment #3: Three people A, B and C play a game such that the order in which they finish is determined purely by chance. Outcome required: B beats A.

9) For each of these experiments below; (a) List all the possible outcomes for the experiment, showing that the outcomes within the experiment are all equally likely. (b) Highlight on your list/diagram the outcomes required.

i) Experiment: Two die are rolled. Outcome required: The score (i.e. the total obtained) is over 5.

ii) Experiment: A coin is tossed and a dice is rolled. If the coin comes up heads, then the score is the number on the die, but if the coin is tails, then the score is twice the number on the die. Outcome required: The score is over 5.

(c) Hence, for each experiment; (i) & (ii), calculate the required probability.

10) For each question below, draw a new probability space diagram and use that highlight the outcomes required and hence find each of the required probabilities: (a) Two dice are rolled; (and the sum of the two numbers is 6). (b) Two dice are rolled; (and the difference between the two numbers is 2). (c) Two dice are rolled; (the sum is 6 and the difference is more than 2). (d) Two dice are rolled; (a prime appears on one die and the sum is less than 8). (e) Two dice are rolled; primes numbers appear on both, an odd number appears on at least one, the sum is greater than 2 and the difference is less than 5.

1) Two dice are rolled. Using a 2 by 2 tree diagram with branches ‘prime’ and ‘not prime’ and listing all of the possible outcomes, find the probability that: (a) prime numbers appear on both dice, (b) at least one prime number appears, (c) only one prime number appears.

2) Two dice are rolled. Draw a tree diagram with branches ‘even’ and ‘odd’ and list all of the possible outcomes. Hence find the probability that: (a) an even number appears on both dice, (b) an odd number appears on both dice. (c) If we want to find p(an even number greater than 2 appears on both dice), then our existing tree diagram is of no use. Draw a new tree diagram and work out this probability

3) A four-sided spinner has the numbers 1 to 4 marked on it. It is spun twice and the two scores are noted. Using a tree diagram with branches ‘1’, ‘2’, ‘3’ and ‘4’, find the probability that: (a) the total score is prime, (b) the two separate scores are both prime, (c) the product of the scores is prime. Could a simpler tree diagram have been used for part (b). If so, what would the branches be?

4) I have two bags each containing four hyacinth bulbs and know that each contains a pink, a blue, a yellow and a white bulb. I take one bulb at random from each bag, find the probability: (a) the hyacinths will be same colour, (b) the hyacinths will be of different colours.

5) Michael Fish (the famous weatherman) says that, if it rains on a particular day, then

p(next day is rainy) =

. But if it is clear, then p(next day is clear) =

.

Given today (Tuesday) has been rainy and horrible: (a) Draw a tree diagram using branches ‘rainy’ and ‘clear’ for Wednesday and Thursday Note: These are called ‘Dependent Events’ and probabilities are now different on each set of branches (b) Find the probability that, including today, there are more rainy days than clear days over the next three days.

6) There are 3 boys and 5 girls in a maths class. Two pupils are chosen at random to go on a date with each other (yuk!) This might not be so good if they turn out to be the same gender (double yuk!) (a) Draw a tree diagram with outcomes ‘boy’ and ‘girl’ from each node. Note: The probabilities are different on each set of branches. (b): Find the probability that the pupils are of different gender (they’ll still have a rubbish date).

7) A set of four cards, numbered 1, 3, 6, 8 are placed face down on the table. One card is turned over, the number noted and then a different card is turned over and its number is noted. The ‘score’ is the total of these two numbers. (a) Explain why there are 4 branches from the first node, but only 3 branches from the other nodes (b) Draw the tree diagram showing the probabilities along each of the branches and all the outcomes with their probabilities. (c) Find the probability that the numbers on both cards are multiples of 3. (d) Find the probability ‘score’ is 9.

8) The probability of passing the driving test on any given attempt is . What proportion of learner drivers take more than 2 attempts to pass?

1) Keval says he did the homework, but left it at home… His teacher is a bit suspicious, so she decides to give him two questions from the homework and writes down five possible answers for each question. Keval must pick the correct answers. If Keval gets both right, she’ll apologise (gulp!), but if he gets both wrong, he’ll be expelled and his hands will be cut off (yelp!)! She thinks that Keval didn’t actually do the homework and will just pick random answers.

(a) Assuming she is right, find the probabilities that: i) She has to apologise to Keval ii) Keval gets expelled iii) Nothing happens

(b) Even if he hasn’t done the homework, is it realistic to say he’ll pick each answer at random?

(c) What effect would it have on our workings if we remove the assumption that he hasn’t done the homework?

2) In England, it rains a lot! In fact, the probability of a rainy day is

.

(a) Assuming independence, find the probability that only one of the next two days is rainy (b) Do you think it is realistic to assume independence. Explain clearly.

3) Jenny tells me she has two daughters (amazing, since Jenny is only 18) (a) Assuming that equal numbers of boys and girls are born, find the probability that of two babies only one is a girl. (b) Do you think it is reasonable to assume independence?

4) Two shots are fired at a target. The probability that the any shot hits the target is

.

Find the probability that exactly one shot hits the target assuming the second shot is independent of the first

5) When a golfer tees off, the probability that it will land on the green is

if it is a still day,

and

if it is a windy day. A golfer plays three holes (so he has to tee off three times).

(a) Find the probability that all three shots will land on the green, given today is windy. (b) Find the probability that all three shots will miss the green, given today is still. (c) Find the probability that exactly two shots will hit the green, given today is still. (d) Find the probability that at least one shot will miss the green, given today is windy.

6) The proportion of the adult population of a town in employment is 80 percent. 70 percent of the employed adults and 40 percent of the unemployed adults own a car. If a person (Serena) is picked at random, what is the probability: (a) Serena is employed and owns a car, (b) Serena is unemployed and owns a car, (c) Serena owns a car.

7) In a general election, 35 percent of the electorate voted Labour. If three voters are picked at random, find the probability: (a) that they all voted Labour, (b) the majority voted Labour.

8) Jugdish is a sales representative for a crockery company, but is not very good at selling; he knows that only 10% of his calls will result in a sale. The boss gives Jugdish 4 leads to call, but unless he makes a sale, he will be sacked! Assuming independence, find the probability that he keeps his lousy job.