Statistics and Probability Answers

A hospital receives 20% of its COVID-19 vaccine shipments from Ghana and the remainder of its shipments from neighbouring countries. Each shipment contains a very large number of vaccine vials. For Ghana’s shipments, 10% of the vials are ineffective. For the neighbouring countries, 2% of the vials are ineffective. The hospital tests 30 randomly selected vials from a shipment and finds that one is ineffective. What is the probability that the shipment came from Ghana.

The lifetime of a machine is continuous on the interval (0, 40) with probability density function f, where f(t) is proportional to (t + 10)−2, and t is the lifetime in years. Calculate the probability that the lifetime of the machine part is less than 10 years. Hint: Show that f(t) is legitimate and find the proportionality constant.

A hospital receives 20% of its COVID-19 vaccine shipments from Ghana and the remainder of its shipments from neighbouring countries. Each shipment contains a very large number of vaccine vials. For Ghana’s shipments, 10% of the vials are ineffective. For the neighbouring countries, 2% of the vials are ineffective. The hospital tests 30 randomly selected vials from a shipment and finds that one is ineffective. What is the probability that the shipment came from Ghana.

B3. An archer shoots arrows at a circular target where the central portion of the target
inside is called the bull. The archer hits the bull with probability 1/32. Assume that
the archer shoots 96 arrows at the target, and that all shoots are independent
(a) Find the probability mass function of the number of bulls that the archer hits.
(b) Give an approximation for the probability of the archer hitting no more than one bull

B2. A package of 6 fuses are tested where the probability an individual fuse is defective is
0.05. (That is, 5% of all fuses manufactured are defective).
(a) What is the probability one fuse will be defective?
(b) What is the probability at least one fuse will be defective?
(c) What is the probability that more than one fuse will be defective, given that at
least one is defective?

B1. Suppose that 3% of computer chips produced by a certain machine are defective. The
chips are put into packages of 20 chips for distribution to retailers. What is the
probability that a randomly selected package of chips will contain at least 2 defective
chips?

A10. Suppose an airline accepted 12 reservations for a commuter plane with 10 seats. They
know that 7 reservations went to regular commuters who will show up for sure. The
other 5 passengers will show up with a 50% chance, independently of each other.
(a) Find the probability that the flight will be overbooked.
(b) Find the probability that there will be empty seats.

A9. A company prices its hurricane insurance using the following assumptions:
i In any calendar year, there can be at most one hurricane.
ii In any calendar year, the probability of a hurricane is 0.05 .
iii The number of hurricanes in any calendar year is independent of the number of
hurricanes in any other calendar year.
Using the company’s assumptions, calculate the probability that there are fewer than
3 hurricanes in a 20-year period.

A8. A financial analyst has found out that policyholders are four times as likely to file two
claims as to file four claims. If the number of claims filed has a posson distribution,
what is the variance of the number of claims filed.

A7. A recent study shows that the annual cost of maintaining a building in Accra averages
200 Cedis with a variance of 260 Cedis. If a tax of 30% is introduced on all items
associated with the maintenance of building i.e. everything is made 30% more expensive.
Calculate the standard deviation of the annual cost of maintaining a building in Accra