L3 Probability Distributions
Question Bank
An airport shuttle bus leaves the terminal every 15 minutes, always on time. Tane walks out to the shuttle stop at a random time.
Using an appropriate probability distribution model, calculate an estimate for the probability that Tane has to wait more than 9 minutes for the shuttle. State the name and parameter(s) of the distribution you used.
A courier company tells Sione that his parcel will be delivered between 9 am and 1 pm, and that any time in that window is equally likely.
Sione has to go out from 11:30 am until 12:15 pm. Using an appropriate probability distribution model, calculate an estimate for the probability that the parcel arrives while he is out. State the name and parameter(s) of the distribution you used.
A fault on a 40 km rural power line is equally likely to be at any point along the line. Let $X$ be the distance, in km, from the substation at one end of the line to the fault.
(a) Sketch the distribution of the distance from the substation to the fault. Label both axes and show the height of the distribution.
(b) Between 12 km and 18 km from the substation the line crosses a river, where a repair needs a boat. Find the probability that the next fault is on this stretch of the line.
At a school's 3 km cross-country race, no student has ever finished in under 11 minutes, and students who walk the whole way take up to 35 minutes. The most common finishing time is 15 minutes.
Using an appropriate probability distribution model, calculate an estimate for the probability that a randomly chosen student finishes in under 15 minutes. State the name and parameter(s) of the distribution you used.
The maximum temperature, $Xnbsp;°C, on a February day in Blenheim is modelled by a triangular distribution with a minimum of 16 °C, a maximum of 36 °C and a most likely value of 24 °C.
(a) Sketch the model. Label $a$, $b$ and $c$, and show the height of the peak.
(b) Use the model to find the probability that the maximum temperature on a February day is above 24 °C.
Records at a kiwi hatchery show that the incubation time of a kiwi egg ranges from 70 days to 90 days, and that the most common incubation time is 78 days.
Staff ask a vet to check any egg that has not hatched after 85 days. Using an appropriate probability distribution model, calculate an estimate for the probability that a randomly chosen egg is checked by the vet. State the name and parameter(s) of the distribution you used.
The milk tanker collects milk from a dairy farm at a random time between 5 am and 7 am each morning, with every time in that window equally likely.
One morning the tanker has still not arrived at 5:45 am. Given this, find the probability that it arrives before 6:15 am.
At a shearing school, the time a trainee takes to shear one sheep ranges from 2 minutes to 6 minutes. The most common time is 3 minutes.
The instructor's goal is for trainees to shear a sheep in under 4 minutes. Using an appropriate probability distribution model, calculate an estimate for the probability that a trainee shears a randomly chosen sheep in under 4 minutes. State the name and parameter(s) of the distribution you used.
A fishing charter boat measures every snapper its customers keep. Snapper shorter than 30 cm must be returned to the sea. The histogram shows the lengths of the 200 snapper kept this summer.
The skipper models the length, $X$ cm, of a kept snapper with a triangular distribution with $a = 30$, $b = 70$ and $c = 36$.
(a) Use the histogram and the context to justify each of the three parameters.
(b) Use the model to estimate the probability that a kept snapper is longer than 50 cm. Compare your answer with the proportion of the 200 snapper that were longer than 50 cm.
Ana drives to work. On the main road her travel time, $X$ minutes, is modelled by a triangular distribution with $a = 20$, $b = 45$ and $c = 25$. On the hill road her travel time, $Y$ minutes, is equally likely to be any value between 25 and 35 minutes.
(a) Sketch each model, showing its height.
(b) Ana needs to get to work within 30 minutes of leaving home. Which road gives her the better chance? Show your working.
The time, $X$ minutes, to download a large game update is modelled by a triangular distribution with $a = 4$, $b = 30$ and $c = 10$. For a time of $m$ minutes between 10 and 30, the probability that a download takes longer than $m$ minutes is
$ P(X > m) = \frac{(30 - m)^2}{520} $
The game's makers want to tell players: "Half of all downloads finish within ___ minutes." Find the median download time.
A pizza shop's delivery times range from 10 minutes to 50 minutes, and the most likely delivery time is 26 minutes. The shop models the delivery time, $X$ minutes, with a triangular distribution with $a = 10$, $b = 50$ and $c = 26$.
For a time $T$ between 26 and 50 minutes, this model gives the probability that a delivery takes longer than $T$ minutes as
$ P(X > T) = \frac{(50 - T)^2}{960} $
The shop wants to promise a free pizza whenever a delivery takes longer than $T$ minutes, but it can afford free pizzas on at most 5% of deliveries. Find the shortest whole number of minutes, $T$, that the shop could promise.
The time, $X$ minutes, that callers spend on hold to a power company's helpline is modelled by a triangular distribution with $a = 2$, $b = 20$ and $c = 6$.
Kahu has already been on hold for 5 minutes. Given this, find the probability that he is on hold for more than 10 minutes altogether.
The departure delay, $X$ minutes, of an evening flight from Wellington to Auckland (how long after its scheduled time it actually takes off) is modelled by a triangular distribution with $a = 0$, $b = 60$ and $c = 10$.
Passengers connecting to an international flight in Auckland miss their connection if the delay is more than 40 minutes.
(a) One evening the flight still has not taken off 25 minutes after its scheduled time. Given this, find the probability that the connecting passengers miss their connection.
(b) The airline rebooks connecting passengers onto a later flight as soon as the chance of them missing their connection is more than 30%. Should it rebook them now? Explain ONE limitation of the triangular model in this situation.