Probability and the binomial distribution – Year 12 A level maths worksheet with answers to download as PDF

\(A\) and \(B\) are events with \(P(A)=0.5\), \(P(B)=0.4\) and \(P(A\cap B)=0.2\).

1. Find \(P(A\cup B)\). [2 marks]
2. Find \(P(A'\cap B')\). [2 marks]
3. Find \(P(A|B)\). [2 marks]
4. Are \(A\) and \(B\) independent? Justify. [2 marks]

Of 40 students, 22 play football (\(F\)), 18 play tennis (\(T\)) and 9 play both.

1. Draw a Venn diagram and find how many play neither sport. [3 marks]
2. Find the probability that a randomly chosen student plays football only. [2 marks]
3. Find \(P(T|F)\). [2 marks]
4. Find the probability that a student plays exactly one sport. [2 marks]

A bag contains 5 red and 3 blue counters. Two counters are taken at random without replacement.

1. Draw a tree diagram showing the probabilities. [3 marks]
2. Find the probability that both are red. [2 marks]
3. Find the probability that the counters are different colours. [3 marks]
4. Find the probability that at least one is blue. [2 marks]

\(P(X)=0.3\) and \(P(Y)=0.5\).

1. Find \(P(X\cup Y)\) if \(X\) and \(Y\) are mutually exclusive. [2 marks]
2. Find \(P(X\cup Y)\) if \(X\) and \(Y\) are independent. [3 marks]
3. Explain why two events with non-zero probabilities cannot be both mutually exclusive and independent. [2 marks]

\(X\sim B(8,0.3)\).

1. Find \(P(X=2)\). [2 marks]
2. Find \(P(X\le1)\). [3 marks]
3. Find \(P(X\ge3)\). [3 marks]
4. Find \(E(X)\) and \(\text{Var}(X)\). [2 marks]

For each situation decide whether \(X\) can be modelled by a binomial distribution. Give the parameters or explain why not.

1. \(X\) is the number of sixes in 10 rolls of a fair die. [2 marks]
2. \(X\) is the number of red cards in a hand of 5 cards dealt from a pack without replacement. [2 marks]
3. \(X\) is the number of attempts until the first six. [2 marks]
4. State the four conditions for a binomial model. [2 marks]

On any day in a particular town the probability of rain is 0.3, independently of other days.

1. Find the probability of no rain for three days in a row. [2 marks]
2. Find the probability of exactly 2 rainy days in 5 days. [3 marks]
3. Find the probability of at least one rainy day in a week. [3 marks]
4. Find the expected number of dry days in a 30-day month. [2 marks]

A factory makes light bulbs. 4% of bulbs are faulty, independently. A box contains 20 bulbs.

1. Define a suitable binomial model. [2 marks]
2. Find the probability that a box has no faulty bulbs. [2 marks]
3. Find the probability that a box has at most 2 faulty bulbs. [4 marks]
4. The factory refunds if more than 2 bulbs are faulty. Find the probability of a refund and the expected number of faulty bulbs per box. [3 marks]

2% of a population have a disease. A test gives a positive result for 95% of people with the disease and for 3% of people without it.

1. Draw a tree diagram. [3 marks]
2. Find the probability that a randomly chosen person tests positive. [3 marks]
3. Given that a person tests positive, find the probability that they have the disease. [3 marks]
4. Explain why the answer to part (c) is surprisingly low. [2 marks]

A multiple-choice test has 12 questions, each with 4 options of which one is correct. Zoe guesses every answer.

1. State the distribution of the number of correct answers. [1 mark]
2. Find the probability that she gets exactly 3 correct. [2 marks]
3. The pass mark is 6. Find the probability that she passes by guessing. [4 marks]
4. Find her expected score and comment. [2 marks]

The probability that Maya\'s bus is late on any day is 0.2, independently of other days. She takes the bus on 5 school days.

1. Find the probability that the bus is never late. [2 marks]
2. Find the probability that it is late exactly twice. [2 marks]
3. Find the probability that it is late on at least 3 days. [4 marks]
4. The school sends a letter if the bus is late on 3 or more days in a week. About how many letters would be sent in a 38-week school year? [2 marks]

A basketball player scores each free throw with probability 0.7, independently. She takes 10 shots.

1. Find the probability she scores exactly 8. [2 marks]
2. Find the probability she scores 8 or more. [3 marks]
3. Find the probability she scores 6 or fewer. [2 marks]
4. Find the mean and standard deviation of the number of baskets. [2 marks]

Two fair dice are rolled and the scores are added.

1. Find the probability that the total is 7. [2 marks]
2. Find the probability that at least one die shows a 6. [2 marks]
3. Given that at least one die shows a 6, find the probability that the total is at least 10. [3 marks]
4. Are the events "at least one six" and "total at least 10" independent? [3 marks]

A drawer holds 6 red and 4 green socks. Three socks are taken at random without replacement.

1. Find the probability that all three are red. [3 marks]
2. Find the probability that exactly two are green. [4 marks]
3. Find the probability that at least one is green. [2 marks]
4. Explain why the number of green socks is not binomial. [1 mark]

150 sixth-form students were asked how they travel to school. The results are shown.

1. Find \(P(\text{bus})\) and \(P(\text{Year 13 and car})\). [2 marks]
2. Find \(P(\text{car}|\text{Year 13})\). [2 marks]
3. Find \(P(\text{Year 12}|\text{walk})\). [2 marks]
4. Test whether "Year 13" and "car" are independent. [4 marks]

Each time an app is opened there is a probability of 0.1 that a new advert is shown. Opens are independent.

1. Show that \(P(\text{at least one})=1-0.9^n\) and check the entry for \(n=20\). [3 marks]
2. Find the smallest \(n\) such that this probability exceeds 0.99. [4 marks]
3. Verify with \(n=43\) and \(n=44\). [2 marks]
4. Find the expected number of adverts in 44 opens. [1 mark]

Three machines \(A\), \(B\) and \(C\) make 50%, 30% and 20% of a factory\'s output. Their defect rates are 1%, 2% and 4%.

1. Find the probability that an item chosen at random is defective. [3 marks]
2. Given that an item is defective, find the probability it came from machine \(C\). [3 marks]
3. Find the probability that two items chosen at random are both defective. [2 marks]
4. How many defectives are expected in a batch of 5 000 items? [1 mark]

\(X\sim B(15,0.4)\). Some cumulative probabilities \(P(X\le r)\) are given.

1. Use the table to find \(P(X\le4)\) and \(P(X=6)\). [3 marks]
2. Find \(P(X\ge8)\). [2 marks]
3. Find the smallest value \(c\) such that \(P(X\le c)>0.5\). [2 marks]
4. Find the mean and standard deviation, and comment on how the mean compares with the value from part (c). [3 marks]

In a club of 60 members, 34 swim (\(S\)), 28 cycle (\(C\)) and 20 run (\(R\)). 12 swim and cycle, 9 swim and run, 8 cycle and run, and 5 do all three.

1. Find the number who do none of the sports. [4 marks]
2. Find the number who do exactly one sport. [4 marks]
3. Find the probability that a randomly chosen member cycles given they run. [2 marks]
4. Find the probability that a member does exactly two sports. [2 marks]

\(X\sim B(n,p)\) has mean 6 and variance 4.2.

1. Find \(n\) and \(p\). [4 marks]
2. Find \(P(X=6)\). [2 marks]
3. Show that \(\dfrac{P(X=r+1)}{P(X=r)}=\dfrac{(n-r)p}{(r+1)q}\) and use it to show that 6 is the most likely value. [5 marks]