Probability: experiments, Venn and tree diagrams – Year 11 GCSE maths worksheet with answers to download as PDF

A fair spinner has eight equal sections numbered 1 to 8.

1. Find the probability of spinning a prime number. [2 marks]
2. Find the probability of spinning a multiple of 3. [2 marks]
3. Find the probability of not spinning a 5. [1 mark]
4. The spinner is spun 96 times. How many times would you expect to spin a 5? [2 marks]

Jamal drops a drawing pin and records how often it lands point up.

1. Work out the relative frequency for each number of drops, to 2 decimal places. [3 marks]
2. Which gives the best estimate of the probability? Explain. [2 marks]
3. Use it to estimate how many times the pin lands point up in 500 drops. [2 marks]

Two fair dice are thrown and the scores are added.

1. Complete the sample space diagram (a 6 by 6 table of totals). [3 marks]
2. Find the probability that the total is 7. [2 marks]
3. Find the probability that the total is at least 10. [2 marks]
4. Find the probability that both dice show the same number. [1 mark]

30 pupils were asked if they play football (\(F\)) or hockey (\(H\)). 12 play football, 10 play hockey and 4 play both.

1. Draw a Venn diagram showing the number in each region. [3 marks]
2. Find \(P(F\cap H)\) and \(P(F\cup H)\). [3 marks]
3. Find the probability that a pupil plays neither sport. [2 marks]

A spinner lands on red with probability 0.3. It is spun twice.

1. Draw a tree diagram and label all branches. [3 marks]
2. Find the probability of two reds. [2 marks]
3. Find the probability of exactly one red. [3 marks]
4. Find the probability of at least one red. [2 marks]

There are 200 pupils in Year 11: 120 are girls and 80 are boys. Three quarters of the girls and half of the boys cycle to school.

1. Complete a frequency tree. [4 marks]
2. How many pupils cycle? [1 mark]
3. A pupil who cycles is picked at random. What is the probability that the pupil is a girl? [2 marks]

A die is rolled 60 times. The results are shown.

1. How many times would you expect each score if the die is fair? [1 mark]
2. Work out the relative frequency of a 6 to 3 decimal places. [2 marks]
3. Is there strong evidence that the die is biased? Explain. [3 marks]
4. How could you get a more reliable result? [1 mark]

At a school, each pupil chooses pizza with probability 0.4, pasta with probability 0.35 and salad with probability 0.25.

1. Check that the probabilities are valid. [1 mark]
2. 240 pupils have lunch. How many are expected to choose pizza? [2 marks]
3. Two pupils choose independently. Find the probability that both choose pasta. [2 marks]
4. Find the probability that neither of them chooses salad. [2 marks]

A bag contains 3 red and 5 blue counters. A counter is taken, its colour is noted and it is put back. Then a second counter is taken.

1. Draw a tree diagram. [3 marks]
2. Find \(P(\text{both red})\) and \(P(\text{both blue})\). [3 marks]
3. Find the probability that the two counters are different colours. [3 marks]

A bag contains 4 red and 6 blue counters. Two counters are taken at random without replacement.

1. Complete a tree diagram. [3 marks]
2. Find \(P(\text{different colours})\). [3 marks]
3. Find \(P(\text{same colour})\). [3 marks]

80 students were asked if they study French (\(F\)) or Spanish (\(S\)). 45 study French, 35 study Spanish and 12 study both.

1. Draw a Venn diagram. [3 marks]
2. Find \(P(F\cup S)\). [2 marks]
3. Find the probability that a student studies exactly one of the two languages. [2 marks]
4. A student who studies French is chosen. What is the probability that they also study Spanish? [2 marks]

Spinner A has the numbers 1, 2, 3, 4 and spinner B has the numbers 1, 2, 3. Both are fair. The two scores are multiplied.

1. Draw a sample space diagram of all the products. [3 marks]
2. Find the probability that the product is even. [2 marks]
3. Find the probability that the product is 6. [2 marks]
4. Find the probability that the product is greater than 5. [2 marks]

The probability of rain on Saturday is 0.4 and on Sunday is 0.3. The days are independent.

1. Draw a tree diagram. [3 marks]
2. Find the probability that it rains on both days. [2 marks]
3. Find the probability that it rains on neither day. [2 marks]
4. Find the probability that it rains on exactly one day. [3 marks]

A footballer scores a penalty with probability 0.8 each time. She takes three penalties independently.

1. Find the probability that she scores all three. [2 marks]
2. Find the probability that she misses at least one. [2 marks]
3. Find the probability that she scores exactly two. [3 marks]
4. Find the probability that she scores none. [1 mark]

50 pupils were asked about music and drama. 28 do music, 22 do drama and 8 do neither. Let \(x\) be the number who do both.

1. Show that \(28-x+x+22-x+8=50\) and solve for \(x\). [4 marks]
2. Complete the Venn diagram. [2 marks]
3. Find the probability that a pupil does exactly one activity. [2 marks]
4. A pupil who does drama is chosen. Find the probability that they also do music. [2 marks]

There are 250 cars in a car park. 60% are petrol and the rest are diesel or electric. 30% of the petrol cars are more than 5 years old and 20% of the other cars are more than 5 years old.

1. Complete a frequency tree. [4 marks]
2. How many cars are more than 5 years old? [2 marks]
3. A car older than 5 years is chosen. Find the probability that it is petrol. [2 marks]
4. Is a randomly chosen car more likely to be an old petrol car or a new non-petrol car? [2 marks]

A bag has 5 counters, of which \(n\) are red. A counter is taken, replaced and a second counter taken. The probability of two reds is 0.16.

1. Write an equation using \(n\). [2 marks]
2. Solve it to find \(n\). [3 marks]
3. Find the probability that the two counters are different colours. [3 marks]

A coin is tossed 400 times and lands on heads 232 times.

1. Work out the relative frequency of heads. [1 mark]
2. How many heads would you expect if the coin is fair? [1 mark]
3. Compare the two and say whether the coin might be biased. [3 marks]
4. Use the relative frequency to estimate the number of heads in 1 000 tosses. [2 marks]
5. How could you test the coin further? [2 marks]

A bag contains 12 sweets: 5 mint, 4 toffee and 3 fruit. Two sweets are taken at random without replacement.

1. Find the probability that both are the same flavour. [4 marks]
2. Find the probability that at least one is fruit. [3 marks]
3. Find the probability that the second sweet is fruit given that the first was mint. [2 marks]

Three fair coins are tossed.

1. List all eight equally likely outcomes. [2 marks]
2. Find the probability of at least two heads. [2 marks]
3. You are told that at least one coin shows heads. Find the probability that all three show heads. [3 marks]
4. Explain why the answer to (c) is not \(\tfrac18\). [2 marks]