L3 Probability Distributions
Question Bank
The number of typing errors on a page of the first draft of a school magazine can be modelled by a Poisson distribution with a mean of 1.4 errors per page.
Use this Poisson model to calculate an estimate for the probability that a randomly chosen page has exactly 2 typing errors.
A university's IT help desk receives calls at an average rate of 2.6 calls per 10 minutes during opening hours.
(a) Using an appropriate probability distribution model, calculate an estimate for the probability that the help desk receives at most 2 calls in a 10-minute period during opening hours. State the name and parameter(s) of the distribution you used.
(b) Calculate an estimate for the probability that the help desk receives more than 4 calls in a 10-minute period during opening hours.
A council inspection finds an average of 2.5 potholes per 10 km on a rural road.
Using an appropriate probability distribution model, calculate an estimate for the probability that a randomly chosen 2 km stretch of the road has no potholes. State the name and parameter(s) of the distribution you used.
A rural ambulance station has an average of 2.2 call-outs per night shift.
(a) Using an appropriate probability distribution model, calculate an estimate for the probability that there are at least 2 call-outs on a night shift. State the name and parameter(s) of the distribution you used.
(b) Give ONE reason why the use of the Poisson distribution may not be appropriate to model this situation.
During a meteor shower, an observer at Lake Tekapo sees meteors at an average rate of 24 per hour.
Using an appropriate probability distribution model, calculate an estimate for the probability that the observer sees at most 1 meteor in a 5-minute period. State the name and parameter(s) of the distribution you used.
On Saturdays, the number of cars arriving at an automatic car wash in an hour can be modelled by a Poisson distribution with a mean of 6.25 cars.
(a) Calculate the standard deviation of the number of cars arriving in an hour.
(b) Use this Poisson model to calculate an estimate for the probability that the number of cars arriving in an hour is within one standard deviation of the mean.
On Friday evenings, a ride-share app receives requests for rides from one suburb at an average rate of 48 per hour.
Use a suitable probability distribution to estimate the probability that at least 10 requests come from this suburb in a 10-minute period on a Friday evening. State the name and parameter(s) of the distribution you used.
A football team in a club competition scores at least one goal in 80% of its games. The number of goals the team scores in a game can be modelled by a Poisson distribution.
(a) Calculate an estimate for the mean number of goals the team scores per game.
(b) Use this Poisson model to calculate an estimate for the probability that the team scores more than 2 goals in a game.
Volunteers at a kiwi sanctuary listen for kiwi calls at night. At one listening post, calls are heard at an average rate of 3.4 per hour, and the number of calls heard in an hour can be modelled by a Poisson distribution.
(a) Use this Poisson model to calculate an estimate for the probability that fewer than 3 calls are heard in an hour.
(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.
A city hospital's maternity unit has an average of 0.45 births per hour. The number of births in a period of time can be modelled by a Poisson distribution.
(a) Calculate the mean and standard deviation of the number of births in an 8-hour shift.
(b) What is the most likely number of births in an 8-hour shift? Justify your answer by comparing probabilities.
Over the past 20 years, one region of New Zealand has had an average of 2.4 felt earthquakes per month (earthquakes strong enough for people to feel).
(a) Using an appropriate probability distribution model, calculate an estimate for the probability that the region has more than 3 felt earthquakes in a month. State the name and parameter(s) of the distribution you used.
(b) Give TWO reasons why the use of the Poisson distribution may not be appropriate to model this situation.
A tour company runs 2-hour dolphin-watching trips in the Bay of Islands. Passengers get a free trip if no dolphins are seen. The company's records show that at least one pod of dolphins is seen on 87% of its 2-hour trips. The number of pods seen on a trip can be modelled by a Poisson distribution.
The company also runs 3-hour trips. Calculate an estimate for the probability that at least 4 pods are seen on a 3-hour trip.
A school's fashion-design department buys fabric in 50-metre rolls from two suppliers. Its records show that 92% of the rolls from Supplier A have at least one flaw. Supplier B says that its fabric has an average of 0.8 flaws per 10 metres.
The two suppliers charge the same price per metre. Students cut the fabric into 2-metre lengths, and a length with a flaw can't be used.
Assume that, for both suppliers, the number of flaws in a length of fabric has a Poisson distribution.
(a) Calculate an estimate for the mean number of flaws on a 50-metre roll from Supplier A.
(b) Compare the two suppliers' fabric for a 2-metre length. Which supplier should the department buy from? Support your answer with calculations.
A district council's road crew clears slips (small landslides) that block a hill-country road. The crew can deal with up to 3 slips in a month; if there are more, the council has to hire a contractor. Over many years the road has averaged 1.5 slips per month, so the council models the number of slips in a month with a Poisson distribution, $\lambda = 1.5$.
(a) Use this Poisson model to calculate an estimate for the probability that the council has to hire a contractor in a given month.
(b) Most slips happen during heavy rain, and one storm often brings down several slips at once. The council's records show that the number of slips per month has a mean of 1.5 but a variance of 4.5.
Give TWO reasons why the use of the Poisson distribution may not be appropriate to model this situation. Then discuss how the true proportion of months with more than 3 slips is likely to compare with the model's probability in (a), and what this means for the council.