L3 Probability Distributions
Question Bank
Road works on a Canterbury highway hold up traffic with a stop/go sign. A driver who is stopped waits for a time that is equally likely to be anything between 1 minute and 9 minutes.
Using an appropriate probability distribution model, calculate an estimate for the probability that a driver who is stopped waits more than 6 minutes.
State the name and parameter(s) of the distribution you used.
Vehicles arrive at a one-lane bridge on a West Coast highway at random, at an average rate of 2.4 vehicles per 5 minutes.
Using an appropriate probability distribution model, calculate an estimate for the probability that exactly 2 vehicles arrive at the bridge in a 5-minute period.
State the name and parameter(s) of the distribution you used.
An archer at a club hits the gold (the centre of the target) with 35% of her arrows. In a practice round she shoots 12 arrows.
Using an appropriate probability distribution model, calculate an estimate for the probability that exactly 4 of the 12 arrows hit the gold.
State the name and parameter(s) of the distribution you used.
When a car breaks down on a rural road, the time a roadside-assistance patrol takes to arrive ranges from 20 minutes to 80 minutes. The most likely time is 30 minutes.
Using an appropriate probability distribution model, calculate an estimate for the probability that the patrol takes more than 30 minutes to arrive.
State the name and parameter(s) of the distribution you used.
A reaction-time game on a phone shows a red screen that turns green after a random delay. The game is set up so that every delay from 2 seconds to 7 seconds is equally likely.
Using an appropriate probability distribution model, calculate an estimate for the probability that the screen turns green in less than 3.5 seconds.
State the name and parameter(s) of the distribution you used.
Hatchery-raised rainbow trout are released into a lake with one fin clipped, so that anglers can recognise them. In the lake, 30% of the rainbow trout have a clipped fin. Over a weekend, an angler catches 10 rainbow trout from the lake.
Using an appropriate probability distribution model, calculate an estimate for the probability that at least 4 of the 10 trout have a clipped fin.
State the name and parameter(s) of the distribution you used.
A marine studies class counts the cockles inside square frames (quadrats), 50 cm by 50 cm, placed at random on a harbour mudflat. The cockles are scattered at random across the mudflat, at an average of 6.4 cockles per quadrat.
Using an appropriate probability distribution model, calculate an estimate for the probability that a quadrat contains fewer than 4 cockles.
State the name and parameter(s) of the distribution you used.
At a sheepdog trial, the time a dog takes to complete its run ranges from 3 minutes to 15 minutes. The most likely time is 6 minutes.
Using an appropriate probability distribution model, calculate an estimate for the probability that a dog takes more than 12 minutes to complete its run.
State the name and parameter(s) of the distribution you used.
An app on Leilani's phone sends her one "daily challenge" alert each day, at a random time between 7 am and 9 pm. Every time in that window is equally likely. At her school, phones are locked away from 8:45 am to 3:15 pm.
(a) Using an appropriate probability distribution model, calculate an estimate for the probability that, on a school day, the alert arrives while Leilani's phone is locked away.
State the name and parameter(s) of the distribution you used.
(b) Calculate an estimate for the probability that the alert arrives while her phone is locked away on all 5 school days in a week.
At a market garden near Dargaville, the weight of a kūmara is modelled by a normal distribution with mean 310 g and standard deviation 60 g. Kūmara lighter than 230 g are sold as "small" at a lower price.
(a) Find the probability that a randomly chosen kūmara is small.
(b) Two kūmara are chosen at random from the day's harvest. Calculate an estimate for the probability that exactly one of them is small. One of the assumptions needed for this calculation is independence. Justify this assumption in context.
On the gravel of the Otago Central Rail Trail, a cyclist gets punctures at random, at an average rate of 0.8 punctures per 100 km. Ngaio is riding 150 km of the trail, and she carries 2 spare tubes.
Using an appropriate probability distribution model, calculate an estimate for the probability that Ngaio gets more than 2 punctures on her 150 km ride.
State the name and parameter(s) of the distribution you used.
A courier delivers to a rural school every school day. The delivery time is equally likely to be any time between 9:00 am and 1:00 pm. The school office is closed for lunch from 12 noon to 1:00 pm.
(a) Find the probability that, on a given day, the courier arrives while the office is closed for lunch.
(b) Find the probability that, over the 10 school days in a fortnight, the courier arrives during lunch on at least 3 days. State the name and parameter(s) of the distribution you used.
(c) On how many of the 10 school days in a fortnight, on average, does the courier arrive during lunch?
A survey found that 18% of New Zealand secondary school students have a phone with a cracked screen. A random sample of 25 secondary school students is chosen.
(a) Using an appropriate probability distribution model, calculate an estimate for the probability that more than 5 of the 25 students have a phone with a cracked screen.
State the name and parameter(s) of the distribution you used.
(b) Give TWO reasons why the use of the probability distribution model selected in part (a) is appropriate.
At a theme park, the time a person spends in the queue for the most popular ride ranges from 5 minutes to 65 minutes. The most likely time is 15 minutes.
Using an appropriate probability distribution model, calculate an estimate for the probability that a person spends less than 45 minutes in the queue.
State the name and parameter(s) of the distribution you used.
The time an installer takes to fit a heat pump is modelled by a triangular distribution with a minimum of 2 hours, a maximum of 7 hours and a most likely time of 3 hours. A job that takes longer than 5.5 hours runs into overtime.
(a) Find the probability that a randomly chosen job runs into overtime.
(b) The installer has three jobs booked this week, at different houses on different days. Calculate an estimate for the probability that at least one of the three jobs runs into overtime. One of the assumptions needed for this calculation is independence. Justify this assumption in context.
Students at a large school sit a 60-minute online practice test and submit it when they finish. No student finishes in under 20 minutes, every student has finished by the end of the hour, and the most likely finishing time is 50 minutes.
(a) Using an appropriate probability distribution model, calculate an estimate for the probability that a student finishes the test in less than 35 minutes.
State the name and parameter(s) of the distribution you used.
(b) Give TWO reasons why the use of the probability distribution model selected in part (a) is appropriate.
At a café, the time from ordering until a flat white is ready ranges from 1 minute to 9 minutes, and the most likely time is 3 minutes.
Toasted sandwiches are made by a different staff member. The time from ordering until a toasted sandwich is ready can be modelled by a normal distribution with mean 5 minutes and standard deviation 0.8 minutes.
Priya orders a flat white and a toasted sandwich at the same time.
(a) Using appropriate probability distribution models, calculate an estimate for the probability that both are ready within 6 minutes. State the name and parameter(s) of each distribution you used.
(b) One of the assumptions needed for the calculation in part (a) is independence. Justify this assumption in context.
When tickets for a popular concert go on sale online, fans who joined the online waiting room early are let in to buy tickets in a random order. A fan's waiting time after the sale opens is equally likely to be anything from 0 to 45 minutes.
Tia joined the waiting room early. Twenty minutes after the sale opens, she is still waiting to be let in.
Using an appropriate probability distribution model, calculate an estimate for the probability that Tia is let in within the next 10 minutes.
State the name and parameter(s) of the distribution you used.
The number of electric cars arriving at a fast-charging station in a 30-minute period is modelled by a Poisson distribution with a mean of 1.8. A 30-minute period in which more than 3 cars arrive is a "busy" period, because some drivers have to queue.
(a) Find the probability that a 30-minute period is busy.
(b) Between 8 am and 2 pm there are 12 half-hour periods. Find the probability that 3 or more of them are busy. State the name and parameter(s) of the distribution you used.
(c) On average, how many of the 12 half-hour periods between 8 am and 2 pm are busy?
Sione is a Year 13 long jumper. The distances of his jumps are modelled by a normal distribution with mean 6.20 m and standard deviation 0.30 m. A jump of at least 6.40 m qualifies him for a national competition.
(a) Find the probability that one of Sione's jumps qualifies.
(b) Find the least number of jumps Sione needs so that the probability that at least one of them qualifies is at least 0.9. State an assumption you made in your calculation, and discuss whether it is likely to be valid in this context.
A school group is walking a coastal track over 5 days in March. In this area, it rains on 25% of days in March.
(a) Using an appropriate probability distribution model, calculate an estimate for the probability that it does not rain on any of the 5 days.
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, and discuss why this is the case. Explain how this could affect your estimate in (a).
In a hospital emergency department, a patient with chest pain needs a blood test result and an X-ray report before a doctor can decide whether they can go home.
Both are ordered when the patient arrives. The time until the blood test result is back can be anywhere between 30 and 80 minutes. The time until the X-ray report is back can be modelled by a normal distribution with mean 50 minutes and standard deviation 12 minutes.
(a) Using appropriate probability distribution models, calculate an estimate for the probability that at least one of the two results takes more than 70 minutes to come back. State the name and parameter(s) of each distribution you used.
(b) State an assumption you made in calculating this probability, and discuss whether it is likely to be valid in this context.
At a little blue penguin colony on the Otago coast, the penguins come ashore after dark. On summer evenings, they come ashore at an average rate of 8 penguins per 10 minutes.
(a) Using an appropriate probability distribution model, calculate an estimate for the probability that at least 12 penguins come ashore in a 10-minute period.
State the name and parameter(s) of the distribution you used.
(b) Penguins often gather in small groups at sea before they come ashore.
Give TWO reasons why the use of the probability distribution model selected in part (a) may not be appropriate to model this situation.
A TV fishing show wants to film at least one "bumper" trip: a half-day charter on which 3 or more kingfish are caught. The number of kingfish caught on a half-day trip is modelled by a Poisson distribution.
Charter A charges $\unicode{36}1400$ per trip and catches a mean of 1.7 kingfish per trip. Charter B charges $\unicode{36}2900$ per trip and catches a mean of 2.6 kingfish per trip. The show must book, and pay for, all of its trips in advance.
(a) Find the probability that a trip with Charter A is a bumper trip.
(b) Show that 8 Charter A trips are not enough, but 9 trips are, for the probability of at least one bumper trip to be at least 0.9.
(c) Charter B needs 4 trips for the probability of at least one bumper trip to be at least 0.9. Which charter should the show use to get a probability of at least 0.9 of filming a bumper trip at the lower total cost? Support your answer with calculations. State an assumption your comparison relies on, and discuss whether it is likely to be valid in this context.
A beekeeper has 12 hives at one site, where the bees feed mainly on mānuka flowers. The amount of honey one hive produces in a season is modelled by a triangular distribution with a minimum of 6 kg, a maximum of 46 kg and a most likely amount of 16 kg.
(a) Calculate an estimate for the probability that more than half of the 12 hives each produce more than 25 kg of honey this season. State the name and parameter(s) of each distribution you used.
(b) State an assumption you made in calculating this probability, and discuss whether it is likely to be valid for these 12 hives.
At a Central Otago vineyard, the lowest temperature on an October night is modelled by a normal distribution with mean 2.5 °C and standard deviation 2.5 °C. On a night when the temperature drops below 0 °C (a frost night), the vineyard must run its frost fans to protect the new vine shoots.
(a) Find the probability that a randomly chosen October night is a frost night.
(b) Assuming nights are independent, find the probability that three nights in a row are all frost nights.
(c) The vineyard manager uses your answer to (b) to say that three frost nights in a row is so unlikely that the vineyard doesn't need to plan for it. Discuss whether the independence assumption you made in (b) is likely to be valid in this context, and describe how violating it could affect the accuracy of your estimate in (b). Comment on the manager's plan.