Probability, Statistics, and Reliability Theory
Practice Questions on Module 1
1) Let A be the event of getting an odd number and B be the event of getting a prime
number in a single throw of a die. What will be the probability that it is either an odd
number or a prime number?
2) A card is drawn from a well-shuffled deck of 52 cards. What is the probability that it is
either a spade or a king?
3) The probabilities that a student will receive an A, B, C or D grade are 0.30, 0.35, 0.20
and 0.15 respectively. What is the probability that a student will receive at least a B
grade?
4) An urn contains 8 white balls and 2 green balls. A sample of three balls is selected at
random. What is the probability that the sample contains at least one green ball?
5) Two balls are drawn at random with replacement from a bag containing 5 blue and 10
red balls. Find the probability that both the balls are either blue or red.
6) Arun and Tarun appear for an interview for two vacancies. The probability of Arun's
selection is 1/3 and that of Tarun's selection is 1/5. Find the probability that
(a) both of them will be selected. (b) none of them is selected.
(c) at least one of them is selected. (d) only one of them is selected.
7) Assume that a certain school contains equal number of female and male students. 5 % of
the male population is football players. Find the probability that a randomly selected
student is a football player male.
8) From a box containing 4 white balls, 3 yellow balls and 1 green ball, two balls are drawn
one at a time without replacement. Find the probability that one white and one yellow
ball is drawn.
9) The odds against A speaking the truth are 3:2 and the odds against B speaking the truth
are 5 : 3. In what percentage of cases are they likely to contradict each other on a
identical issue?
10) An urn contains 6 red balls and 3 blue balls. One ball is selected at random and is
replaced by a ball of the other color. A second ball is then chosen. What is the
conditional probability that the first ball selected is red, given that the second ball was
red?
11) A family has five children. Assuming that the probability of a girl on each birth was 0.5
and that the five births were independent, what is the probability the family has at least
one girl, given that they have at least one boy?
12) A box contains 2 green and 3 white balls. A ball is selected at random from the box. If
the ball is green, a card is drawn from a deck of 52 cards. If the ball is white, a card is
drawn from the deck consisting of just the 16 pictures. (a) What is the probability of
drawing a king? (b) What is the probability of a white ball was selected given that a king
was drawn?
13) Suppose box A contains 4 red and 5 blue chips and box B contains 6 red and 3 blue
chips. A chip is chosen at random from the box A and placed in box B. Finally, a chip is
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Probability, Statistics, and Reliability Theory
chosen at random from among those now in boxes B. What is the probability a blue chip
was transferred from box A to box B given that the chip chosen from box B is red?
14) Sixty percent of new drivers have had driver education. During their first year, new
drivers without driver education have probability 0.08 of having an accident, but new
drivers with driver education have only a 0.05 probability of an accident. What is the
probability a new driver has had driver education, given that the driver has had no
accident the first year?
15) One-half percent of the population has AIDS. There is a test to detect AIDS. A positive
test result is supposed to mean that you have AIDS but the test is not perfect. For people
with AIDS, the test misses the diagnosis 2% of the times. And for the people without
AIDS, the test incorrectly tells 3% of them that they have AIDS. (a) What is the
probability that a person picked at random will test positive? (b) What is the probability
that you have AIDS given that your test comes back positive?
16) A card is drawn at random from an ordinary deck of 52 cards and replaced. This is done
a total of 5 independent times. What is the conditional probability of drawing the ace of
spades exactly 4 times, given that this ace is drawn at least 4 times?
17) An urn contains 6 red balls and 3 blue balls. One ball is selected at random and is
replaced by a ball of the other color. A second ball is then chosen. What is the
conditional probability that the first ball selected is red, given that the second ball was
red?
18) Suppose box A contains 4 red and 5 blue chips and box B contains 6 red and 3 blue
chips. A chip is chosen at random from the box A and placed in box B. Finally, a chip is
chosen at random from among those now in boxes B. What is the probability a blue chip
was transferred from box A to box B given that the chip chosen from box B is red?
19) Each coefficient in the equation 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0can take the values 1,2,3,4 or 5. Find
the probability that the roots of the equation are imaginary.
20) There are seven letters and seven addressed envelopes. If the letters are placed in the
envelopes at random, find the probability that none of the letters is in the correct
envelope.
21) The chance that a doctor will diagnose a disease, say D, correctly is 60%. The chance
that a patient will die by his treatment after correct diagnosis is 40% and the chance of
death by wrong diagnosis is 70%. The patient of the doctor, who has disease D, died.
What is the chance that is disease was diagnosed correctly?
22) In the test an examiner either guesses or copies or knows the correct answer. The
probability that he makes a guess is 1/3 and the probability that he copies the answer is
1/6. The probability that his answer is correct, given that he copied it is 1/3 and the
probability that his answer is correct, given that he guessed it is 1/4. Find the probability
that he knew that answer, given that he correctly answered it.
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