Sample 2 Exam
1. Eleven people are going on a skiing trip in 3 cars that hold 2, 4, and 5 passengers, respectively.
a) In how many ways is possible to transport the 11 people to the ski lodge?
b) In how many ways is possible to transport the 11 people, if two friends want to be together in any of the cars?
c) In how many ways is possible to transport the 11 people, if two of them don’t want to be together in the same
car?
2. An automobile dealer has three mechanics, A, B, and C, each of whom handles dealer preparation for one-third of
the new cars. Among the tasks they perform. is that of filling up the windshield washer container with washer fluid. It
is known that A forgets to do this 6% of the time, B forgets 9% of the time, and C forgets 15% of the time. Suppose
you pick up your new car at the dealership.
a) What is the probability that it has no washer fluid?
b) What is the probability that if it has washer fluid, C prepared your car?
3. The length of time between two breakdowns of a particular electrical generator is best approximated by an
exponential distribution with mean equal to 10 days.
a) What is the probability that the generator which broke down today, will break down again within the next 14
days?
b) What is the probability that the generator will operate for more than 20 days without a breakdown?
c) What is the probability in a month of 30 days at this generator at most two breakdowns will occur (Poisson
distribution)?
4. Consider the following density function of a random variable X.
a) Find the cumulative (distribution) function of X.
b) Calculate the expected value.
c) Calculate the probability of the given event: 25.269.1 XP .
5. A machine fills cans with cola. The amount of fill is a random variable with a mean of 12 ounces and standard
deviation of 0.20 ounces. What is the probability that the quantity of cola in the can will be at most 11.50 or at least
12.50 ounces if
a) the distribution of X is unknown;
b) X is uniformly distributed;
c) X has normal distribution?
9938.05.2;9772.02;9332.05.1;8413.01;6915.05.0
6. X and Y are two independent random variables having the following marginal probabilities:
Y \ X -1 0 2 P(Y)
-2 0.5
0 0.3
1 0.2
P(X) 0.1 0.3 0.6
a) Calculate the joint distribution.
b) Find the expected values of X and Y.
c) Calculate the standard deviations of X and Y.
c) Find the value of E(X∙Y).