L3 Probability Distributions
Question Bank
About 9% of blood donors have O negative blood. At a donor session, 40 people give blood.
Calculate the mean and standard deviation of the number of donors at the session who have O negative blood. State the name and parameter(s) of the distribution you used.
About 22% of the students at a large school walk to school. A random sample of 15 students is chosen.
(a) Using an appropriate probability distribution model, calculate an estimate for the probability that exactly 3 of the 15 students walk to school. State the name and parameter(s) of the distribution you used.
(b) Calculate an estimate for the probability that fewer than 2 of the 15 students walk to school.
A basketball player makes 80% of her free throws. In a tournament she takes 25 free throws.
Using an appropriate probability distribution model, calculate an estimate for the probability that she makes more than 20 of them. State the name and parameter(s) of the distribution you used.
A quiz has 12 multiple-choice questions, each with four options. Only one option is correct for each question. A student who has not studied guesses every answer at random.
(a) Using an appropriate probability distribution model, calculate the probability that the student gets exactly 3 answers correct. State the name and parameter(s) of the distribution you used.
(b) To pass, a student needs at least 6 correct answers. Calculate the probability that the student passes.
Before lambing, a farmer has her ewes scanned. From past seasons, each ewe has a 0.55 probability of carrying twins. A mob of 20 ewes is scanned.
(a) Using an appropriate probability distribution model, calculate an estimate for the probability that no more than 12 of the 20 ewes are carrying twins. State the name and parameter(s) of the distribution you used.
(b) Give ONE reason why the use of the probability distribution model selected in part (a) is appropriate.
At a vehicle testing station, 30% of cars fail their warrant of fitness (WoF) inspection. One morning, 14 cars are inspected.
Using an appropriate probability distribution model, calculate an estimate for the probability that between 3 and 6 of these cars (inclusive) fail. State the name and parameter(s) of the distribution you used.
At a kiwifruit packhouse, 6% of fruit is usually rejected for export because of marks or poor shape. An inspector takes a random sample of 50 kiwifruit from one grower's bins and finds 8 that must be rejected.
(a) Would finding 8 or more rejects in a random sample of 50 be unusual, if the grower's fruit is rejected at the usual 6%? Using an appropriate probability distribution model, calculate a probability to support your answer, and state the name and parameter(s) of the distribution you used. Explain what the result suggests about this grower's fruit.
(b) Give TWO reasons why the use of the probability distribution model selected in part (a) is appropriate.
A farmer is planting native seedlings along a stream. She is choosing between two plans that cost about the same:
Plan A: plant 40 seedlings with protective sleeves. Each has a 0.9 probability of surviving its first year.
Plan B: plant 60 seedlings without sleeves. Each has a 0.6 probability of surviving its first year.
(a) Show that both plans give the same expected number of surviving seedlings.
(b) Which plan gives a more consistent number of surviving seedlings? Support your answer with calculations.
A seed company says that 85% of its pea seeds germinate. Mia plants a tray of 20 of these seeds.
(a) Using an appropriate probability distribution model, calculate an estimate for the probability that at least 18 of the seeds germinate. State the name and parameter(s) of the distribution you used.
(b) To apply this distribution, at least one assumption must be made. Identify ONE assumption that may be invalid for Mia's tray of seeds and discuss why this is the case.
At a supermarket's self-checkouts, each customer has the same probability of needing help from a staff member, independently of the other customers. The probability that none of the next 20 customers needs help is 0.1887.
(a) Find the probability that a customer needs help.
(b) Using an appropriate probability distribution model, calculate an estimate for the probability that more than 3 of the next 20 customers need help. State the name and parameter(s) of the distribution you used.
A bakery makes 8 gluten-free pies each morning. Assume that 36 customers buy a pie each morning, and that each one chooses gluten-free with probability 0.15, independently of the others.
(a) Using an appropriate probability distribution model, calculate an estimate for the probability that, on a given morning, at least one customer who wants a gluten-free pie misses out. State the name and parameter(s) of the distribution you used.
(b) The owner calls a morning "badly planned" if a customer misses out on a gluten-free pie, or if 6 or more gluten-free pies are left over. Calculate an estimate for the probability that a morning is badly planned, and explain what this probability means for the bakery.
Conservation workers set up a trail camera to check whether a stoat is living on a small island. If a stoat is there, the camera has a 0.15 probability of photographing it on any one night, independently of other nights.
Find the smallest number of nights the camera must run so that, if a stoat is there, the probability that it is photographed at least once is at least 0.95.
An airline sells 75 tickets for a flight with 72 seats, because on average 6% of ticket holders do not turn up. This flight is popular with families and sports teams, who usually travel together.
(a) Using an appropriate probability distribution model, calculate an estimate for the probability that more than 72 ticket holders turn up for the flight. State the name and parameter(s) of the distribution you used.
(b) To apply this distribution, at least one assumption must be made. Identify ONE assumption that may be invalid for this flight and discuss why this is the case. How could this affect the probability you found in (a)?
A food delivery company claims that 95% of its orders arrive on time. Sam has ordered from the company 20 times this year, and only 16 of these orders arrived on time.
(a) Suppose the company's claim is true. Using an appropriate probability distribution model, calculate an estimate for the probability that 16 or fewer of Sam's 20 orders arrive on time. State the name and parameter(s) of the distribution you used.
(b) Does Sam's experience give good reason to doubt the company's claim? To apply the distribution in (a), at least one assumption must be made: in your answer, identify ONE assumption that may be invalid for Sam's orders and discuss why this is the case.