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Suppose it is known that the weight of a certain population of individuals are approximately normally distributed with a mean 80 kg and a standard deviation of 16 kg. What is the probability that a person picked at random from this group will weight between 60 and 110 kg?
Suppose that all athletes run 200 metres and the time they take to run is normally distributed with mean 12 seconds and a standard deviation of 3 seconds. The coach has decided that 40 percent of the athletes who can run the distance in the least time will be sent to participate in the Olympics. What is the cutoff score that will be decide which members of the time will qualify.
You can’t stand your probability professor. The amount of time you can sit in her classroom before storming out is modelled by an exponential random variable, where on average you storm out after 10 minutes.
(a) Given that you’ve already been sitting in her classroom for 30 minutes, what is the probability you will still be sitting in the classroom for 5 more minutes?
(b) What is the probability you are going to storm out in the next 10 minutes?
Suppose that the time between customer arrivals in a store is given by an exponential random variable X ∼ Exp(λ), such that the average time between arrivals is 2 minutes. Suppose you walk past the store and notice it’s empty. What is the probability from the time you walk past the store, the store remains empty for more than 5 minutes?
The length of time patients must wait to see a doctor at an emergency room of a large hospital is uniformly distributed between 40 minutes and 3 hours. What is the probability that a patient will have to wait between 50 minutes and 1.5 hours to see a doctor?
According to the Insurance Institute of America, a family of four spends between Ghc 400 and Ghc 3,800 per year on all types of insurance. Suppose the money spent is uniformly distributed between these amounts.
(a) What is the mean amount spent on insurance?
(b) What is the standard deviation of the amount spent?
(c) If we select a family at random, what is the probability they spend less than Ghc 2,000 per year on insurance per year?
(d) If we select a family at random, what is the probability they spend less than Ghc 2,000 per year on insurance per year? (e) What is the probability a family spends more than Ghc 3,000 per year?
After your complaint about their service, a representative of an insurance company promised to call you “between 7 and 9 this evening.” Assume that this means that the time T of the call is uniformly distributed in the specified interval.
(a) Compute the probability that the call arrives between 8:00 and 8:20.
(b) At 8:30, the call still hasn’t arrived. What is the probability that it arrives in the next 10 minutes?
(c) Assume that you know in advance that the call will last exactly 1 hour. From 9 to 9:30, there is a game show on TV that you wanted to watch. Let M be the amount of time of the show that you miss because of the call. Compute the expected value of M.
The number of people arriving for treatment at an emergency room can be modelled by a Poisson process with a mean of five people per hour.
(a) What is the probability that exactly four arrivals occur at a particular hour?
(b) What is the probability that at least four people arrive during a particular hour?
(c) How many people do you expect to arrive during a 45-min period?
An oil company conducts a geological study that indicates that an exploratory oil well should have a 20% chance of striking oil.
(a) What is the probability that the first strike comes on the third well drilled?
(b) What is the probability that the third strike comes on the seventh well drilled?
(c) What is the mean and variance of the number of wells that must be drilled if the oil company wants to set up three producing wells?
A pipe-smoking mathematician always carries two boxes of matches - one is his right pocket and another in his left pocket. Each time he needs a match, he is equally likely to take it from either pocket. Suppose that each box initially contains n matches. What is the probability that once the mathematician discovers that one of the boxes is empty, there are exactly k matches in the other box, k = 0,1,...n?
