Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 4.17—poisson Distribution with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 4.17—poisson Distribution and mirrors Paper 1, 2, 3 style where relevant.
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A logistics company tracks the number of its heavy-duty trucks that require unscheduled maintenance each day. Based on historical data, the number of such trucks per day can be modeled by a Poisson distribution with a mean of 2.4.
Calculate the probability that on a given day, exactly 3 heavy-duty trucks require unscheduled maintenance.
Find the probability that exactly 10 heavy-duty trucks require unscheduled maintenance over a 5-day work week.
Find the probability that over a four-day period, unscheduled maintenance is required on the first two days only.
Over 30 days, find the probability that exactly 5 days have no unscheduled maintenance.
The company expands its fleet by adding medium-duty trucks. The number of medium-duty trucks requiring maintenance per day also follows a Poisson distribution, with a mean of maintenance requests per truck per day. Maintenance events for all trucks are assumed to be independent.
Determine the minimum total number of trucks in the fleet (the existing heavy-duty fleet plus medium-duty trucks) such that the probability of having more than 15 maintenance requests in a single day is at least 0.15.
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.
The number of occurrences in each unit interval is assumed to follow a Poisson distribution with mean 4.0. A researcher records results from 100 unit intervals and uses a goodness of fit test at the significance level to check this model. The observed frequencies are: 0 (10), 1 (15), 2 (22), 3 (21), 4 (14), 5 (9), 6 or more (9).
Calculate the expected frequencies for each category under the Poisson model.
Carry out a goodness-of-fit test (state df and conclusion).
If the mean is 4.8, find the probability that the total number of events over 100 unit intervals is at least 460, using a normal approximation. Include a labelled sketch of the normal curve showing the continuity correction point and the shaded tail.
A local post office models the number of customers arriving per -minute period with a Poisson distribution of mean .
Find the probability that exactly customers arrive in a -minute period.
Calculate the probability that at least customers arrive over a -minute period.
In a -hour period, find the probability that exactly two of the six -minute periods have fewer than customers.
A new counter is opened, increasing the mean to customers per minutes. Using a normal approximation, find the probability that more than customers arrive in a -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 . 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.