Exercise 1: (2 points)

A large assembly plant for motor vehicles recently detected two

serious manufacturing faults in some of its models. It is estimated that

3% the percentage that a car chosen at random has the first fault,

at 5% the one she presents the second and at 2% the one she presents both.

By choosing a car at random in this factory, what is the probability

she presents

1. at least one of these faults?

2. at least one of these faults but not both together?

Exercise 2:

The engineers of a

They must continually check for corrosion inside

pipes that are part of the cooling systems. The interior state of

pipes cannot be observed directly, but non-destructive testing may

wind give an indication of corrosion. The test is not perfect. The test has

0.7 probability of detecting internal corrosion when corrosion

internal is present but it also has a probability of 0.1 that corrosion

internal detection, when there is no internal corrosion. Suppose the

probability that the pipe has internal corrosion is 0.15. If the test detects the

internal corrosion, what is the probability that there is actually corrosion

internal?

Exercise 3:

Ottawa Confederation Line Trains Arrive During Rush Hours

at a station according to a Poisson process, averaging 10 trains per hour.

1. What is the distribution of the waiting time of three trains? What is the

probability that this waiting time is at most 25 minutes? What

are the mean and variance of this random variable?

2. No train arrives during the first 5 minutes of waiting. Which is

the probability that there will be no train in the next 3 minutes?

3. What is the probability that two trains will arrive in twelve minutes?

4. A

95%, at least one train arrives in t-minutes. What is approximate-

t in minutes?

5. A period of 5 minutes is said to be empty if it contains no arrivals

of trains. What is the probability that at most 5 minutes

be empty in an hour?

Exercise 4:

The probability law of a couple of discrete random variables (X; Y) is

given by:

X/Y -1 -1

-1 1/10 3/10

-1 5/10 1/10

1. Find the marginal laws of X and Y.

2. Are the variables X and Y independent?

3. We consider the random variable U = l X- Y l .

(a) Determine the probability law of U.

(b) Determine the distribution function of U.

Exercise 5:

Suppose you play a series of ping pong games with Clemonell

until one of you wins six. Suppose that the probability

that Cl emonell wins is 0.58 and that the games are independent.

1. What is the probability that the series will end in the ninth game?

2. If the series is in the ninth game, what is the probability that you

prevail?

Exercise 6:

Suppose mountaineering expeditions fail after three events

mutually exclusive such as: the probability that an exp edition will fail due to

its unfavorable weather and weather conditions is 0.6; the one she fails in

reason for climber injury is 0.25; finally, the probability that the failure is due

insufficient or lost equipment is 0.15. During a recent summer, four

expeditions failed in their attempt to climb Mount Logan, the point

culmination of Canada.

1. What is the probability that two failures are due to bad weather and

the other two have equipment failures?

2. What is the probability that two failures are due to bad weather and

that at least one failure is caused by insufficient equipment?

3. What is the probability that exactly two failures will be caused by

unfavorable weather and weather conditions?

Exercise 7:

The daily amount of water drunk by an elephant in Serengeti National Park

is uniformly distributed between 0 and 60 liters.

1. What is the probability that an elephant will drink only 25 liters of water

during a day?

2. What is the probability that it will take 40 days for an elephant

drink at least 45 liters of water for the first time?

3. What is the probability that it will take 11 days for an elephant

drink at least 45 liters of water for the fifth time?

4. We observe a certain parasite in these elephants. We have identified in

average 35 of these parasites per el ephant. What is the probability that

find more than 2 parasites on an elephant chosen at random?

Exercise 8:

A dealer has 45 cars, 25 of which are compact. What

is the probability that over the next 10 sales:

1. 8 are compact?

2. Is a probabilistic approximation possible under these conditions?

Exercise 9:

A company manufactures fuses whose operating life can be

approached by a normal distribution having an average of 1000 hours and

a standard deviation of 200 hours.

1. What is the probability that a fuse will work for more than 1330

hours?

2. The company would like to offer a replacement warranty for everything

fuse whose operating time does not exceed x0 hours. D eterminer

x0 if the company intends to replace a maximum of 2.5% of its fuses

with this guarantee.

3. A sample of 10 fuses is taken at random and put into service in a

how they work independently of each other. Calculate

the probability that at least 8 of the fuses in the sample will not work

more after 1330 hours?

Exercise 10:

Let X be a uniform random variable over the interval [-1; 3] such as:

y^2 + 4Xy + 1 = 0:

1. What is the probability that this equation has complex roots?

2. What is the probability that this equation has real roots

separate?