IB Mathematics Applications & Interpretation SL 4.8—binomial Distribution Questionbank

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 $0.78$, and the probability that the member uses the cardio area is $0.45$. Assume that on each visit, members use at least one of the two areas.

On a typical day, $150$ members visit the gym.

It is found that $65\%$ of the members visiting the gym are male, and that $88\%$ of these male members use the weights area.

Assume different members' choices of areas are independent. The remaining $35\%$ of members are female.

(a) Find the probability that a randomly chosen member uses both the weights and cardio areas.

(b) Find the probability that a randomly chosen member only uses the cardio area.

(c) On a typical day, find the expected number of members to use the weights area.

(d) On a typical day, find the probability that more than $75$ members use the cardio area.

A theme park has two main attraction areas: rollercoasters and water rides. The probability that a randomly observed guest uses the rollercoaster area is $0.68$, and the probability that the guest uses the water rides area is $0.55$. Assume that each guest uses at least one of the two areas.

On a typical day, $140$ guests visit the theme park.

It is found that $58\%$ of the guests are teenagers, and that $80\%$ of these teenage guests use the rollercoaster area.

Assume different guests' choices of attraction areas are independent.

(a) Find the probability that a randomly chosen guest uses both the rollercoaster and water rides areas.

(b) Find the probability that a randomly chosen guest only uses the water rides area.

(c) On a typical day, find the expected number of guests who use the rollercoaster area.

(d) On a typical day, find the probability that more than $85$ guests use the water rides area.

An online quiz bank contains 6 probability questions, 3 calculus questions and $t$ geometry questions. A question is selected at random and then returned to the bank. This is repeated eight times.

(a) Calculate the probability that the first question selected is a probability question.

A concert venue has two popular areas: a merchandise stand and a refreshment stall. The probability that a randomly observed visitor goes to the merchandise stand is $0.56$, and the probability that the visitor goes to the refreshment stall is $0.46$. Assume that each visitor goes to at least one of the two areas.

On a typical evening, $200$ visitors enter the venue.

It is found that $60\%$ of the visitors have seated tickets, and that $70\%$ of these visitors with seated tickets go to the merchandise stand.

(a) Find the probability that a randomly chosen visitor goes to both the merchandise stand and the refreshment stall.

(b) Find the probability that a randomly chosen visitor only goes to the refreshment stall.

(c) On a typical evening, find the expected number of visitors who go to the merchandise stand.

(d) On a typical evening, find the probability that more than $100$ visitors go to the refreshment stall.

(c) Calculate the largest number of reservations the hotel can accept such that the expected number of guests arriving is at most $58$.

(d) Calculate, to the nearest dollar, the expected change in net revenue if the hotel accepts $60$ reservations instead of only $58$.

(b) Let $X$ be the total number of probability questions selected. Determine the minimum value of $t$ for which $E(X)+Var(X)<6$.