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Probability and Statstics - Prob - Theory - problems, Study notes of Economics

Indian Institute of Information Technology (IIIT)Economics

This document about Probability Theory Problems, assignment of probabilities, Examples, probabilities.

Typology: Study notes

2010/2011

Uploaded on 09/03/2011

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Probability Theory Problems
1. Is the following assignment of probabilities is possible? Give the reason.
P(A)=0.35, P(B)=0.25, P(AB)=0.30
2. If P(A)=2P(B)=0.5 and probability that exactly one of the events A or B occurs is
0.35, find the probability that both the events A and B occur.
3. In problem 2, what is the probability that at most one of the events A and B
occur?
4. In a certain state, 25% of cars emit excessive amount of pollutants. If the
probability is 0.95 that a car emitting excessive amount of pollutants will fail the
state's vehicular test, and the probability is 0.15 that a car not emitting the
excessive amount of pollution will fail the test, what is the probability that a car
which fails the test actually emits excessive amount of pollutants.
5. A box contains 6 balls, 2 white and 4 red. Balls are drawn randomly one by one
without replacement until a white ball is obtained. Write down the sample space
and probabilities associated with each elementary event. What is the probability
that number of draws is less than or equal to 3?
6. Let P(A)=a, P(B)=b and . In terms of a, b and c, find the probability that exactly
one of the two events will occur.
7. Suppose that we are concerned with the completion of a highway construction
job, which may be delayed because of a strike. The probability that there will be
a strike is 0.40. The probability that the job will be completed in time if there is
no strike is 0.85 and the probability that the job will be completed on time if there
is a strike is 0.35. What is the probability that the job will be completed on time?
If the job is completed on time, what is the probability that there was no strike?
8. For three events A, B and C, let
Prove that A,B,C are pair-wise independent but not mutually independent. Also
obtain the probability of occurrence of exactly one of the events A,B,C.
9. An explosion at a construction site could have occurred as the result of one of the
four possible reasons, static electricity, malfunctioning of equipment, carelessness
or sabotage. Interviews with construction engineers analyzing the risks involved
led to the estimates that such an explosion would occur with probabilities 0.20 as
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Probability Theory Problems

  1. Is the following assignment of probabilities is possible? Give the reason. P(A)=0.35, P(B)=0.25, P(AB)=0.
  2. If P(A)=2P(B)=0.5 and probability that exactly one of the events A or B occurs is 0.35, find the probability that both the events A and B occur.
  3. In problem 2, what is the probability that at most one of the events A and B occur?
  4. In a certain state, 25% of cars emit excessive amount of pollutants. If the probability is 0.95 that a car emitting excessive amount of pollutants will fail the state's vehicular test, and the probability is 0.15 that a car not emitting the excessive amount of pollution will fail the test, what is the probability that a car which fails the test actually emits excessive amount of pollutants.
  5. A box contains 6 balls, 2 white and 4 red. Balls are drawn randomly one by one without replacement until a white ball is obtained. Write down the sample space and probabilities associated with each elementary event. What is the probability that number of draws is less than or equal to 3?
  6. (^) Let P(A)=a, P(B)=b and. In terms of a, b and c, find the probability that exactly one of the two events will occur.
  7. Suppose that we are concerned with the completion of a highway construction job, which may be delayed because of a strike. The probability that there will be a strike is 0.40. The probability that the job will be completed in time if there is no strike is 0.85 and the probability that the job will be completed on time if there is a strike is 0.35. What is the probability that the job will be completed on time? If the job is completed on time, what is the probability that there was no strike?
  8. For three events A, B and C, let

Prove that A,B,C are pair-wise independent but not mutually independent. Also obtain the probability of occurrence of exactly one of the events A,B,C.

  1. An explosion at a construction site could have occurred as the result of one of the four possible reasons, static electricity, malfunctioning of equipment, carelessness or sabotage. Interviews with construction engineers analyzing the risks involved led to the estimates that such an explosion would occur with probabilities 0.20 as

a result of static electricity, with a probability 0.30 due to malfunctioning of equipment, with probability 0.50 due to carelessness and with probability 0. due to sabotage. It is also felt that the prior probabilities of the four causes of explosion are, respectively, 0.20, 0.40, 0.25, and 0.15. Based on this information a. find the probability of an explosion at the construction site, b. if an explosion has occurred at the construction site, what is the most likely cause of explosion?

  1. If the probabilities are, respectively, 0.23,0.24 and 0.38 that a car stopped at a roadblock will have faulty breaks, badly worn tires, faulty breaks and/or badly worn tires, what is the probability that such a car will have both faulty breaks and badly worn tires.
  2. A biology professor has two graduate assistants helping him with his research. The probability that the older of the two assistants will be absent on any given day is 0.08, the probability that the younger of the two will be absent on a given day is 0.05 and the probability that both of them will be absent on a given day is 0.02. Find the probabilities that i. (^) either or both of the assistants will be absent on a given day; ii. at least one of the assistants will not be absent on a given day; iii. only one of the assistants will be absent on a given day.
  3. A box of fuses contains 20 fuses, of which 5 are defective. If three fuses are selected randomly and removed from the box randomly (without replacement), what are the probabilities that 12.a. All the three are defective? 12.b. At most two are defective? 12.c. At least two are defective?
  4. A finite discrete sample space is consist of the four points denoted {(110),(101),(011),(000)} and each point has probability 1/4. The event A (^) i (I=1,2,3) occurs if there is a 1 at the i-th place. Thus A1={(110),(101). Show that A1, A (^) 2, A 3 are pairwise independent but not mutually independent.
  5. A balanced dice is tossed twice. Let A be the event that an even number comes up in the first toss, B is the event that an even number turns up in the second toss and

20.ii. the distribution function of X; 20.iii. expected number of observed defectives.

  1. If 40% of mice used in an experiment will become aggressive within one minute after having been administered an experimental drug, find the probabilities of following events: 20.iv. Out of 6 such mice, 5 will become aggressive within one minute of administering the drug; 20.v. Out of 10 such mice, at least nine will become aggressive after administering the drug.
  2. If the probability is 0.30 that a child exposed to a certain contagious disease will catch it, what is the probability that the tenth boy exposed to the disease will be the third to catch it.
  3. A certain type of fabric has, on the average, 2 defects per 10 square yards. If we assume a Poisson distribution, what is the probability that a 30 square yard bolt of this fabric will have (i) no defects; (ii) 4 or more defects. Obtain the expected number of defects in a 40 square yards bolt of such fabric.
  4. If the probability is 0.60 that a person will believe a rumor about the transgressions of a certain politician, find the probabilities that: 12.d. out of ten person who hear the rumor, at most two will believe it; 12.e. the eighth person to hear the rumor will be the fifth to believe it;
  5. Records show that the probability is 0.0004 that a car will break down while driving through a certain tunnel. Find the probabilities that 12.f. Among 2000 cars driving through the tunnel at most one will break, 12.g. Among 8,000 cars driving through the tunnel, at least two will break.
  6. The length of time (in minutes) for one individual to be served at a cafeteria is a random variable with the probability density

Find the mean and variance What is the probability that the time for one individual to be served exceeds 4 minutes? What is the probability that out of four

individuals visiting the cafeteria; time to be served for no individual exceeds 4 minutes?

  1. Let Z is a random variable having a standard normal distribution. Using normal table find the probabilities that this random variable will take on a value (a) greater than 1.14, (b) less than -0.36, (c) between -0.46 and -0.09, (d) between -0.58 and 1.12.
  2. Suppose that the amount of cosmic radiation to which a person is exposed when flying by jet across the United States is a random variable having a normal distribution with a mean 4.35 mrem and a standard deviation of 0.59 mrem. What is the probability that a person will be exposed to more than 5.20 mrem of cosmic radiation on such a flight? If a flight carries 100 passengers, what is the probability that no passenger will be exposed to more than 4.35 mrem of radiation? (Use normal table to evaluate the first probabilities. For evaluating the second probability, you don’t require normal table.)
  3. The probability mass function of a random variable X is given by

Find P(X>6X>3).

  1. The life length (in years) of a satellite is a random variable with pdf given by

Determine (i) the mean life of the satellite, (ii) reliability function for the satellite, and failure rate for the satellite.

  1. The mileage which a car owner gets with a certain kind of tire is a random variable following a normal distribution with mean =20,000 miles and standard deviation =200. Find the probability that among 4 car owners, who use this particular kind of tire, tires of all the cars last more than 19,500 miles but less than 20,500 miles. It has been given that
  2. The length of time (in minutes) that a person takes to check and reply his e-mails is a random variable with the probability density function