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.
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.
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 6.4.
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.
The number of events in a unit interval follows a Poisson distribution with mean 6.2. Consider a sequence of 10 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 each interval has at least 3 events.
A Poisson distribution with mean 4.5 models the number of events per unit interval. A study collects data over 100 unit intervals to test if the data fits this distribution using a goodness of fit test at a significance level. The observed frequencies are: 0 (8), 1 (18), or more (5).
Calculate the expected frequencies for each category under the Poisson model.
Perform the goodness of fit test, stating the degrees of freedom and conclusion.
If the mean is actually 5.0, find the probability that the total number of events over 100 unit intervals is at least 480 , using a normal approximation. Include a sketch.
The number of delivery trucks arriving at a warehouse per hour follows a Poisson distribution with a mean of 3.7.
Find the probability that no trucks arrive in a given hour.
Calculate the probability that exactly 8 trucks arrive over a 2 -hour period.
Find the probability that at least one truck arrives in each of three consecutive hours.
A coffee kiosk models the number of orders placed per 2-minute period with a Poisson distribution of mean 1.6.
Find the probability that exactly 2 orders are placed in a 2 -minute period.
Calculate the probability that at least 3 orders are placed over a 6-minute period.
Find the probability that no orders are placed in one 2 -minute period, given that at least one order is placed in the previous 2 -minute period.
The number of events in a fixed interval is modeled by a Poisson distribution with mean 3.6 per unit interval. A researcher conducts a hypothesis test to determine if a new condition has increased the mean rate of events, using a significance level. The test is based on observing 120 events over a 30 -unit interval.
State the null and alternative hypotheses for the test.
Assuming the null hypothesis is true, state the distribution of the number of events observed and find the critical value for the test.
Calculate the probability of a Type II error if the true mean rate is 4.2 per unit interval.
Using a normal approximation, find the probability that the total number of events in a 50 -unit interval exceeds 200 under the new condition (mean 4.2). Include a sketch of the normal distribution with the relevant area shaded.
A nature trail records the number of bird sightings per day, modeled by a Poisson distribution with mean 5.8.
Find the probability of exactly 7 sightings in a day.
Over a 4-day period, calculate the probability of at least 20 sightings.
Find the expected number of days in a 20 -day period with exactly 4 sightings.
Due to increased bird activity, the mean rises to 7.3. Find the least number of days required for the probability of at least 40 sightings to exceed 0.98 , using a normal approximation. Include a sketch.
A railway station models the number of passengers boarding a train per minute with a Poisson distribution of mean 3.9.
Find the probability that exactly 2 passengers board in a minute.
Calculate the probability that at least 10 passengers board over a 5 -minute period.
Determine the expected number of minutes in a 30 -minute period with exactly 1 passenger boarding.
If a new train service increases the mean to 5.1 per minute, find the probability that fewer than 25 passengers board in a 6 -minute period, using a normal approximation. Include a sketch.