L3 Probability Distributions
Question Bank
A survey of households in a small Waikato town gives this probability distribution for $X$, the number of cars owned by a household.
| $x$ | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| $P(X = x)$ | 0.05 | 0.31 | 0.42 | 0.15 | 0.07 |
How many cars, on average, does a household in this town own?
The table shows the probability distribution of $X$, the number of goals a school's First XI football team scores in a game.
| $x$ | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| $P(X = x)$ | 0.20 | 0.35 | $k$ | 0.12 | 0.05 |
(a) Find the value of $k$.
(b) Find the probability that the team scores more than 2 goals in a game.
A survey asked New Zealand households how many paid video-streaming services they subscribe to. The bar graph shows the probability distribution of $X$, the number of services a household subscribes to. No household had more than four, but the bar for three services is missing.
(a) Find the probability that a household subscribes to exactly 3 streaming services.
(b) Calculate the mean number of streaming services per household.
A whale-watching company in Kaikōura records the number of sperm whales seen on each trip. The table shows the probability distribution of $X$, the number of sperm whales seen on a trip.
| $x$ | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| $P(X = x)$ | 0.06 | 0.38 | 0.33 | 0.17 | 0.06 |
(a) The company's website says that on more than half of its trips, passengers see at least two sperm whales. Is this claim supported by the distribution? Show your working.
(b) Calculate the mean number of sperm whales seen on a trip, and explain what this value means in this context.
A netball club sells 500 raffle tickets at $\unicode{36}10$ each. The prizes are one prize of $\unicode{36}1500$, two prizes of $\unicode{36}250$ and five prizes of $\unicode{36}100$. Each ticket can win at most one prize.
Let $X$ be the amount of money, in dollars, won by one ticket.
(a) Write down, as a table, the probability distribution of the amount won by one ticket.
(b) Calculate the mean amount won per ticket, and use it to find the expected gain or loss for a person who buys one ticket.
A dental clinic records $X$, the number of new fillings a patient needs at a routine check-up. The probability distribution of $X$ is shown below.
| $x$ | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| $P(X = x)$ | 0.52 | 0.27 | 0.15 | 0.06 |
Calculate the mean and standard deviation of the number of new fillings a patient needs at a check-up.
A taxi company records $X$, the number of passengers on each trip. Its taxis carry at most four passengers. The probability distribution of $X$ is shown below.
| $x$ | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| $P(X = x)$ | 0.52 | 0.30 | 0.12 | 0.06 |
An airport shuttle company's vans carry up to 11 passengers. Its records show that the number of passengers on a shuttle trip has a standard deviation of 2.3.
Show that the number of passengers on a taxi trip has a smaller standard deviation than the number on a shuttle trip, and give ONE reason, in context, why this might be the case.
Each time a student takes out books from a school library, the number of books, $X$, is recorded. The probability distribution of $X$ is shown below, where $a$ and $b$ are unknown.
| $x$ | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| $P(X = x)$ | 0.25 | $a$ | $b$ | 0.15 |
The mean number of books taken out each time is 2.3.
(a) Find the values of $a$ and $b$.
(b) Calculate the standard deviation of the number of books taken out each time.
An electrician charges a call-out fee of $\unicode{36}75$ plus $\unicode{36}90$ for each hour, or part of an hour, spent on a job. Let $X$ be the number of hours charged for a job. The probability distribution of $X$ is shown below.
| $x$ | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| $P(X = x)$ | 0.25 | 0.45 | 0.20 | 0.10 |
(a) Calculate the mean and standard deviation of the number of hours charged for a job.
(b) Calculate the mean and standard deviation of the total charge for a job.
At a pet clinic's Saturday vaccination morning, one vet vaccinates dogs and another vet, in a separate room, vaccinates cats. Let $X$ be the number of dogs and $Y$ the number of cats vaccinated in a 15-minute period. The distribution of $X$ is shown below.
| $x$ | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| $P(X = x)$ | 0.10 | 0.25 | 0.35 | 0.20 | 0.10 |
The number of cats vaccinated in a 15-minute period has a mean of 1.2 and a standard deviation of 0.9. Assume that the numbers of dogs and cats vaccinated are independent.
(a) Calculate the mean and standard deviation of the number of dogs vaccinated in a 15-minute period.
(b) Calculate the mean and standard deviation of the total number of animals vaccinated in a 15-minute period.
A kayak-hire business at Kaiteriteri charges $\unicode{36}25$ for the first hour and $\unicode{36}15$ for each extra hour, but no hire costs more than $\unicode{36}60$. Kayaks are hired for 1, 2, 3 or 4 hours. Let $X$ be the number of hours a kayak is hired for. The probability distribution of $X$ is shown below.
| $x$ | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| $P(X = x)$ | 0.20 | 0.35 | 0.25 | 0.20 |
(a) How many hours, on average, is a kayak hired for?
(b) The manager estimates the mean charge per hire as $25 + 15\,(E(X) - 1)$ dollars, where $E(X)$ is the mean hire time from (a). Explain why this gives the wrong answer, and calculate the correct mean charge per hire.
Tama runs a small online business and is choosing between two courier companies, A and B. Let $X$ be the number of working days a parcel takes to arrive. The table gives the probability distribution of $X$ for each courier, from its records.
| $x$ | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| A | 0.10 | 0.40 | 0.40 | 0.08 | 0.02 |
| B | 0.40 | 0.25 | 0.10 | 0.10 | 0.15 |
Tama tells his customers when to expect their parcels, so he most wants delivery times that are predictable. He would also like parcels to arrive quickly.
Calculate the mean and standard deviation of the number of working days a parcel takes to arrive with each courier, and use them to recommend a courier for Tama.
A busy Wellington restaurant records $X$, the number of bookings that don't turn up ("no-shows") on a Saturday night. The table gives the probability distribution of $X$ before the restaurant began sending text-message reminders the day before each booking, and after.
| $x$ | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Before | 0.10 | 0.25 | 0.30 | 0.20 | 0.15 |
| After | 0.40 | 0.40 | 0.15 | 0.04 | 0.01 |
(a) Calculate the mean and standard deviation of the number of no-shows on a Saturday night before the reminders.
(b) Without calculating the mean or standard deviation after the reminders, explain what effect the reminders have had on the expected value and on the standard deviation of the number of no-shows on a Saturday night. Explain what the changes mean for the restaurant.
Two ferry companies, A and B, sail across Cook Strait. Let $X$ be the number of days in a winter week on which Company A cancels at least one sailing, and $Y$ the number of days in the same week on which Company B does. The distribution of $X$ is shown below.
| $x$ | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| $P(X = x)$ | 0.30 | 0.25 | 0.20 | 0.15 | 0.10 |
Records also show that the standard deviation of $Y$ is 1.1 days, and the standard deviation of $X + Y$ (the two companies' counts added together) is 2.3 days.
(a) Calculate the standard deviation of the number of days in a winter week on which Company A cancels at least one sailing.
(b) Are the two companies' numbers of days with cancellations, $X$ and $Y$, independent? Support your answer with statistical calculations, and explain what your answer suggests about the two companies' cancellations.