Practice IB Mathematics Applications & Interpretation (AI) Topic SL 4.8—binomial Distribution with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 4.8—binomial Distribution and mirrors Paper 1, 2, 3 style where relevant.
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A boutique hotel has a capacity of rooms. Hotel records show that, on average, of guests with reservations arrive to check in on a given night. Assuming this pattern remains unchanged, the hotel overbooks, relying on the fact that no more than guests will arrive. The number of guests who arrive follows a binomial distribution with success probability . Reservations cost each. If more guests arrive than rooms available, the hotel pays to relocate each extra guest.
Suppose the hotel accepts reservations. Determine the probability that the number of guests arriving is greater than .
State the mean number of guests that will arrive if the hotel accepts exactly reservations.
Calculate the largest number of reservations the hotel can accept such that the expected number of guests arriving is at most .
Calculate, to the nearest dollar, the expected change in total revenue if the hotel accepts reservations instead of only .
A university gym has two areas: a weights area and a cardio area. The probability that a randomly observed member uses the weights area is , and the probability that the member uses the cardio area is . Assume that on each visit, members use at least one of the two areas.
On a typical day, members visit the gym.
It is found that of the members visiting the gym are male, and that of these male members use the weights area.
Find the probability that a randomly chosen member uses both the weights and cardio areas.
Find the probability that a randomly chosen member only uses the cardio area.
On a typical day, find the expected number of members to use the weights area.
On a typical day, find the probability that more than members use the cardio area.
A gym member is chosen at random. Find the probability that the member is male and uses the weights area.
A female member is chosen at random. Find the probability that the member uses the weights area.
Trays at a warehouse each contain five ceramic mugs. Each mug is either intact or cracked. It is hypothesized that the number of cracked mugs in a tray follows a binomial distribution. A sample of 120 trays was inspected with the following results.
| Number of cracked mugs per tray | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Frequency | 22 | 43 | 34 | 16 | 4 | 1 |
Find the arithmetic mean number of cracked mugs per tray.
Thus, determine an estimate for , the success probability of choosing a cracked mug.
Using a goodness of fit test, evaluate at the 5% significance level if the binomial model is an appropriate fit for this dataset.
A theme park has two main attraction areas: rollercoasters and water rides. The probability that a randomly observed guest uses the rollercoaster area is , and the probability that the guest uses the water rides area is . Assume that each guest uses at least one of the two areas.
On a typical day, guests visit the theme park.
It is found that of the guests are teenagers, and that of these teenage guests use the rollercoaster area.
Find the probability that a randomly chosen guest uses both the rollercoaster and water rides areas.
Find the probability that a randomly chosen guest only uses the water rides area.
On a typical day, find the expected number of guests who use the rollercoaster area.
On a typical day, find the probability that more than guests use the water rides area.
A theme park guest is chosen at random. Find the probability that the guest is a teenager and uses the rollercoaster area.
A non-teenage guest is chosen at random. Find the probability that the guest uses the rollercoaster area.
An online quiz bank contains 6 probability questions, 3 calculus questions and geometry questions. A question is selected at random and then returned to the bank. This is repeated eight times.
Calculate the probability that the first question selected is a probability question.
Let be the total number of probability questions selected. Determine the minimum value of for which .