Exercises on Counting and Assigning Probabilities
Final answers shown between brackets

a. In how many ways can we choose 2 workers and 5 engineers out of 5 workers and 8 engineers? [560]
b. In the question above, if 2 engineers and 2 workers are close friends, what is the probability that they will be all chosen? [0.036]

A small manufacturing company will start operating a night shift. There are 20 machinists employed by the company.
a) If a night crew consists of three machinists, how many different crews are possible? [1140]
b) If the machinists are ranked 1, 2, ... ,20 in order of competence, how many of these crews would not have the best machinist?[969]
c) How many of the crews would have at least one of the 10 best machinists? [1020]
d) If one of these crews is selected at random to work on a particular night, what is the probability that the best machinist will not work that night? [0.85]

A mathematics professor wishes to schedule an appointment with each of her eight teaching assistants, four male and four female, to discuss her calculus course. Suppose all possible orderings of appointments are equally likely to be selected.

a) What is the probability that one female assistant is among the first three the professor meets with? [0.429]
b) What is the probability that at least one female assistant is among the first three the professor meets with? [0.929]
c) What is the probability that after the first five appointments she has met with all female assistants? [0.0714]
d) Suppose the professor has the same eight assistants the following semester and again schedules appointments without regard to the ordering during the first semester. What is the probability that the orderings of appointments are different? [0.9999752]

A student has a box containing 25 computer disks, of which 15 are blank and the other 10 are not. If she randomly selects disks one by one, what is the probability that at least two must be selected to find one that is blank? [0.40]

There are 25 students in a classroom. What is the probability that at least two of them have the same birthday (day and month)? Assume that the year has 365 days. State your assumptions.

Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 60% have an emergency locator, whereas 90% of the aircraft not discovered do not have such a locator. Suppose a light aircraft has disappeared.

a) If it has an emergency locator, what is the probability that it will not be discovered? [0.067]
b) If it does not have an emergency locator, what is the probability that it will be discovered? [0.509]

A friend who works in a big city owns two cars, one small and one large. Three-quarters of the time he drives the small car to work, and one-quarter of the time he takes the large car. If he takes the small car, he usually has little trouble parking, and so is at work on time with probability 0.9. If he takes the large car, he is on time to work with probability 0.6. Given that he was on time on a particular morning, what is the probability that he drove the small car? [0.82]

Two methods, A and B, are available for teaching a certain industrial skill. The failure rate is 20% for A and 10% for B. However, B is more expensive and hence is used only 30% of the time. (A is used the other 70%) A worker is taught the skill by one of the methods, but fails to learn it correctly. What is the probability that he was taught by Method A? [0.823]

A diagnostic test for a certain disease is said to be 90% accurate in that, if a person has the disease, the test will detect it with probability 0.9. Also, if a person does not have the disease, the test will report that he doesn't have it with probability 0.9. Only 1% of the population has the disease in question. If a person is chosen at random from the population, and the diagnostic test reports him to have the disease, what is the conditional probability that he does, in fact, have the disease? Are you surprised by the size of the answer? Would you call this diagnostic test reliable? (S&M, 2.37) [0.0833]

An assembler of electric fans uses motors from two sources. Company A supplies 90% of the motors and company B supplies the other 10%. Suppose it is known that 5% of the motors supplied by company A are defective and 3% of the motors supplied by company B are defective. An assembled fan is found to have a defective motor. What is the probability that this motor was supplied by company B? [0.0625]

It is known that 20% of the residents of Kuwait shop at Khalydia Cooperative market ONLY, while 30% shop at Adyliah Cooperative market ONLY. It is also known that 60% shop at either Khalydia or Adyliah. Using K to represent shopping at Khalydia and A to represent shopping at Adyliah, find:

a) the probability that a customer will shop at both Khalydia and Adyliah. [0.1]
b) the probability that a customer will not shop at either Khalydia or Adyliah. [0.4]
c) the probability that a customer will shop at Khalydia given that he shopped at Adyliah. [0.25]
d) the probability that a customer will shop in either Khalydia or Adyliah but not in both. [0.5]

If a point is selected at random inside a circle, what is the probability that the point is closer to the center than to the circumference? [0.25]

Consider forming 3-letter passwords using the five letters A, B, C, D, and E, where each letter may be repeated as wanted.

a) How many passwords can be formed? [125]
b) What is the probability that a randomly chosen password will have 3 different letters? [0.48]
c) What is the probability that a randomly chosen password will consist of the same letter repeated three times?
  [0.04]
d) What is the probability that a randomly chosen password will have one letter repeated twice? [0.48]