Practice AHL 4.17—Poisson distribution with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.
A call center receives an average of 5 calls per hour. The number of calls received follows a Poisson distribution.
Calculate the probability that the call center receives exactly 3 calls in a given hour.
Find the probability that the call center receives more than 7 calls in a given hour.
Hank sets up a bird table in his garden to provide the local birds with some food. Hank notices that a specific bird, a large magpie, visits several times per month and he names him Bill. Hank models the number of times per month that Bill visits his garden as a Poisson distribution with mean 3.1.
Using Hank's model, find the probability that Bill visits the garden on exactly four occasions during one particular month.
Over the course of 3 consecutive months, find the probability that Bill visits the garden on exactly 12 occasions.
Over the course of 3 consecutive months, find the probability that Bill visits the garden during the first and third month only.
Find the probability that over a 12-month period, there will be exactly 3 months when Bill does not visit the garden.
After the first year, a number of baby magpies start to visit Hank's garden. It may be assumed that each of these baby magpies visits the garden randomly and independently, and that the number of times each baby magpie visits the garden per month is modelled by a Poisson distribution with mean 2.1.
Determine the least number of magpies required, including Bill, in order that the probability of Hank's garden having at least 30 magpie visits per month is greater than 0.2.
The random variable , representing the number of occurrences in a unit interval, follows a Poisson distribution with mean 7.8.
Find the probability that exactly 6 occurrences are observed in a unit interval.
Calculate the probability that the number of occurrences in a 2-unit interval is between 12 and 16 inclusive.
Determine the smallest integer such that the probability of at least occurrences in a 3-unit interval is less than 0.01.
The number of marathons that Audrey runs in any given year can be modelled by a Poisson distribution with mean .
Calculate the probability that Audrey will run at least two marathons in a particular year.
Find the probability that she will run at least two marathons in exactly four out of the following five years.
Two independent random variables and follow Poisson distributions.
Given that and , calculate:
.
.
.
A call center models the number of inquiries received per 5-minute period with a Poisson distribution of mean 4.1.
Find the probability that exactly 3 inquiries are received in a 5-minute period.
Over a 20-minute period, calculate the probability of at least 18 inquiries.
Determine the probability that exactly one 5-minute period in a 30-minute interval has no inquiries.
If a new system reduces the mean to 3.2 per 5 minutes, find the least number of 5-minute periods required for the probability of at least 30 inquiries to exceed 0.85, using a normal approximation. Include a sketch.
A local post office models the number of customers arriving per 10-minute period with a Poisson distribution of mean .
Find the probability that exactly 5 customers arrive in a 10-minute period.
Calculate the probability that at least 15 customers arrive over a 30-minute period.
In a 1-hour period, find the probability that exactly two 10-minute periods have fewer than 3 customers.
A new counter is opened, increasing the mean to 8.2 customers per 10 minutes. Using a normal approximation, find the probability that more than 50 customers arrive in a 1-hour period. Include a sketch of the normal distribution with the relevant area shaded.
A Chocolate Shop advertises free gifts to customers that collect three vouchers. The vouchers are placed at random into 10% of all chocolate bars sold at this shop. Kati buys some of these bars and she opens them one at a time to see if they contain a voucher. Let $\text{P}(X = n)$ be the probability that Kati obtains her third voucher on the $n$th bar opened.
(It is assumed that the probability that a chocolate bar contains a voucher stays at 10% throughout the question.)
It is given that $\text{P}(X = n) = \frac{n^2 + an + b}{2000} \times 0.9^{n - 3}$ for $n \geqslant 3, n \in \mathbb{N}$.
Kati’s mother goes to the shop and buys $x$ chocolate bars. She takes the bars home for Kati to open.
Show that $\text{P}(X = 3) = 0.001$ and $\text{P}(X = 4) = 0.0027$.
Find the values of the constants $a$ and $b$.
Deduce that $\frac{\text{P}(X = n)}{\text{P}(X = n - 1)} = \frac{0.9(n - 1)}{n - 3}$ for $n > 3$.
(i) Hence show that $X$ has two modes $m_1$ and $m_2$.
(ii) State the values of $m_1$ and $m_2$.
Determine the minimum value of $x$ such that the probability Kati receives at least one free gift is greater than 0.5.
A bookstore sells an average of 2 rare books per day. The number of rare books sold follows a Poisson distribution.
Determine the probability that the bookstore sells no rare books in a day.
Calculate the probability that the bookstore sells at least 3 rare books in a day.
The number of events in a unit interval follows a Poisson distribution with mean . Consider a sequence of independent unit intervals.
Find the probability that exactly 4 events occur in a unit interval.
Calculate the probability that exactly three intervals have exactly 5 events.
Determine the probability that the total number of events across the 10 intervals is at least 70, given that exactly 6 events occurred in the first interval.
Practice AHL 4.17—Poisson distribution with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.
A call center receives an average of 5 calls per hour. The number of calls received follows a Poisson distribution.
Calculate the probability that the call center receives exactly 3 calls in a given hour.
Find the probability that the call center receives more than 7 calls in a given hour.
Hank sets up a bird table in his garden to provide the local birds with some food. Hank notices that a specific bird, a large magpie, visits several times per month and he names him Bill. Hank models the number of times per month that Bill visits his garden as a Poisson distribution with mean 3.1.
Using Hank's model, find the probability that Bill visits the garden on exactly four occasions during one particular month.
Over the course of 3 consecutive months, find the probability that Bill visits the garden on exactly 12 occasions.
Over the course of 3 consecutive months, find the probability that Bill visits the garden during the first and third month only.
Find the probability that over a 12-month period, there will be exactly 3 months when Bill does not visit the garden.
After the first year, a number of baby magpies start to visit Hank's garden. It may be assumed that each of these baby magpies visits the garden randomly and independently, and that the number of times each baby magpie visits the garden per month is modelled by a Poisson distribution with mean 2.1.
Determine the least number of magpies required, including Bill, in order that the probability of Hank's garden having at least 30 magpie visits per month is greater than 0.2.
The random variable , representing the number of occurrences in a unit interval, follows a Poisson distribution with mean 7.8.
Find the probability that exactly 6 occurrences are observed in a unit interval.
Calculate the probability that the number of occurrences in a 2-unit interval is between 12 and 16 inclusive.
Determine the smallest integer such that the probability of at least occurrences in a 3-unit interval is less than 0.01.
The number of marathons that Audrey runs in any given year can be modelled by a Poisson distribution with mean .
Calculate the probability that Audrey will run at least two marathons in a particular year.
Find the probability that she will run at least two marathons in exactly four out of the following five years.
Two independent random variables and follow Poisson distributions.
Given that and , calculate:
.
.
.
A call center models the number of inquiries received per 5-minute period with a Poisson distribution of mean 4.1.
Find the probability that exactly 3 inquiries are received in a 5-minute period.
Over a 20-minute period, calculate the probability of at least 18 inquiries.
Determine the probability that exactly one 5-minute period in a 30-minute interval has no inquiries.
If a new system reduces the mean to 3.2 per 5 minutes, find the least number of 5-minute periods required for the probability of at least 30 inquiries to exceed 0.85, using a normal approximation. Include a sketch.
A local post office models the number of customers arriving per 10-minute period with a Poisson distribution of mean .
Find the probability that exactly 5 customers arrive in a 10-minute period.
Calculate the probability that at least 15 customers arrive over a 30-minute period.
In a 1-hour period, find the probability that exactly two 10-minute periods have fewer than 3 customers.
A new counter is opened, increasing the mean to 8.2 customers per 10 minutes. Using a normal approximation, find the probability that more than 50 customers arrive in a 1-hour period. Include a sketch of the normal distribution with the relevant area shaded.
A Chocolate Shop advertises free gifts to customers that collect three vouchers. The vouchers are placed at random into 10% of all chocolate bars sold at this shop. Kati buys some of these bars and she opens them one at a time to see if they contain a voucher. Let $\text{P}(X = n)$ be the probability that Kati obtains her third voucher on the $n$th bar opened.
(It is assumed that the probability that a chocolate bar contains a voucher stays at 10% throughout the question.)
It is given that $\text{P}(X = n) = \frac{n^2 + an + b}{2000} \times 0.9^{n - 3}$ for $n \geqslant 3, n \in \mathbb{N}$.
Kati’s mother goes to the shop and buys $x$ chocolate bars. She takes the bars home for Kati to open.
Show that $\text{P}(X = 3) = 0.001$ and $\text{P}(X = 4) = 0.0027$.
Find the values of the constants $a$ and $b$.
Deduce that $\frac{\text{P}(X = n)}{\text{P}(X = n - 1)} = \frac{0.9(n - 1)}{n - 3}$ for $n > 3$.
(i) Hence show that $X$ has two modes $m_1$ and $m_2$.
(ii) State the values of $m_1$ and $m_2$.
Determine the minimum value of $x$ such that the probability Kati receives at least one free gift is greater than 0.5.
A bookstore sells an average of 2 rare books per day. The number of rare books sold follows a Poisson distribution.
Determine the probability that the bookstore sells no rare books in a day.
Calculate the probability that the bookstore sells at least 3 rare books in a day.
The number of events in a unit interval follows a Poisson distribution with mean . Consider a sequence of independent unit intervals.
Find the probability that exactly 4 events occur in a unit interval.
Calculate the probability that exactly three intervals have exactly 5 events.
Determine the probability that the total number of events across the 10 intervals is at least 70, given that exactly 6 events occurred in the first interval.