Problem 1. Prasad is known to speak the truth 3
times out of 5. He throws a die and reports that it is 1. Find the probability
that it is actually 1.
Problem 2. Pawan speaks the truth 8 times out
of 10 times. A die is tossed. He reports that it was 5. What is the probability
that it was actually 5?
Problem 3. A bag contains 1 white and 6 red
balls, and a second bag contains 4 white and 3 red balls. One of the bags is
picked up at random and a ball is randomly drawn from it, and is found to be
white in colour. Find the probability that the drawn ball was from the first
bag.
Problem 4. There are two bags I and II. Bag I
contains 3 white and 4 black balls and Bag II contains 5 white and 6 black
balls. One ball is drawn at random from one of the bags and is found to be
white. Find the probability that it was drawn from bag I.
Problem 5. There are two bags I and II. Bag I
contains 3 white and 2 red balls and Bag II contains 4 white and 5 red balls.
One ball is drawn at random from one of the bags and is found to be red. Find
the probability that it was drawn from bag II.
Problem 6. A company has two plants to
manufacture motor cycle. 70% motor cycles are manufactured at the first plant,
while 30% are manufactured at the second plant. At the first plant, 80% motor
cycles are rated of the standard quality while at the second plant, 90% are rated
of standard quality. A motor cycle , randomly picked up, is found to be of
standard quality. Find the probability that it has come out from the second
plant.
Problem 7. A company has two plants to
manufacture bicycles. 60% bicycles are manufactured at the first plant, while
40% are manufactured at the second plant. At the first plant, 80% motor cycles
are rated of the standard quality while at the second plant, 90% are rated of
standard quality. A bicycle , randomly picked up, is found to be of standard
quality. Find the probability that it has come out from the second plant.
Problem 8. A insurance company insured 2000 scooters
and 3000 motor cycles. The probability of an accident involving a scooter is
0.01 and that of a motor cycle is 0.02. An insured vehicle met with an
accident. Find the probability that the accident vehicle was a motor cycle.
Problem 9. Three bags contain balls as shown in
the following table: A bag is chosen at random and two balls are drawn. They
happen to be white and red. What is the probability that they came from the
third bag?
|
Bag |
White Balls |
Black Balls |
Red Balls |
|
I |
1 |
2 |
3 |
|
II |
2 |
1 |
1 |
|
III |
4 |
3 |
2 |
Problem 10. Three urns A,B and C contains 6 red and 4 white balls; 2
red and 6 white balls; and 1 red and 5 white balls respectively. An urn is chosen
at random and a ball is drawn. If the ball drawn is found to be red, find the
probability that the ball was drawn from urn A.
Problem 11. A factory has three machines A,B
and C, which produce 100,200 and 300 items of a particular type daily. The
machines produce 2%, 3% and 5% defective items respectively. One day when the
production was over, an item was picked up randomly and found to be defective.
Find the probability that it is manufactured by machine A.
Problem 12. In a bolt factory, machines A,B and
C manufacture 25%,35% and 40% respectively of total bolts. Of their outputs,
5%, 4% and 2% respectively are defective bolts. A bolt is drawn at random and
is found to be defective. Find the probability that it is manufactured by
machine B.
Problem 13. A bag contains 4 red and 4 black
balls, and a second bag contains 2 red and 6 black balls. One of the bags is
picked up at random and a ball is randomly drawn from it, and is found to be
red in colour. Find the probability that the drawn ball was from the first bag.
Problem 14. Probability that Ravi speaks truth
is 4/5 . A coin is tossed, Ravi reports that a head appears. Find the
probability that actually there was head.
Problem 15. There are three coins. One is a two
headed coin, another is a biased coin that comes up heads 75% of the time and
third is an unbiased coin. One of the three coins is chosen at random and
tossed, it shows heads, what is the probability that it was the two headed
coin?
Problem 16. Suppose that 5% of men and 0.25% of
women have grey hair. A grey haired person is selected at random. What is the
probability of this person being male? Assume that there are equal number of
males and females.
Problem 17. Of the students in a college, it is
known that 60% reside in hostel and 40% are day scholars ( not residing in
hostel). Previous year results report that 30% of all students who reside in
hostel attain A grade and 20% of day scholars attain A grade in their
examination. One student is chosen at random and he has an A grade, what is the
probability that the student is from hostel?
Problem 18. A laboratory COVID-19 test is 99%
effective in detecting when it is in fact, present. However, the test also
yields a false positive result for 0.5% of the healthy person tested (i.e., if
a healthy person is tested, then, with probability 0.005, the test will imply
he has the disease). If 0.1% of the population actually has the disease, what
is the probability that a person has the disease given that his test result is
positive?
Problem 19. A card from a pack of 52 cards is
lost. From the remaining cards of the pack, two cards are drawn and are found
to be both diamonds. Find the probability of the lost card being a diamond.
Problem 20. Suppose a girl throws a die. If she
gets a 5 or 6, she tosses a coin three times and notes the number of heads. If
she gets 1,2,3 or 4, she tosses a coin once and notes whether a head or tail is
obtained. If she obtained exactly one head, what is the probability that she
threw 1,2,3 or 4 with the die?
Problem 21. Two groups are competing for the
position on the Board of directors of a corporation. The probabilities that the
first and the second group will win are 0.6 and 0.4 respectively. Further, if
the first group wins, the probability of introducing a new product is 0.7 and
the corresponding probability is 0.3 if the second group wins. Find the
probability that the new product introduced was by the second group.
Problem 22. A manufacturer has three machine
operators A, B and C. The first operator A produces 1% defective items, where
as the other two operators B and C produce 5% and 7% defective items
respectively. A is on the job for 50% of the time, B is on for 30% and C is on
for 20% of the time. A defective item is produced, what is the probability it
was produced by A?
Problem 23. In answering a question on a
multiple choice test, a student either knows the answer or guesses. Let ¾ be
the probability that he knows the answer and ¼ be the probability that he
guesses. Assuming that a student who guesses at the answer will be correct with
probability ¼ . What is the probability that the student knows the answer given
that he answered it correctly?
Problem 24. There are two bags I and II. Bag I
contains 3 white and 4 red balls and Bag II contains 5 white and 6 red balls.
One ball is drawn at random from one of the bags and is found to be red. Find
the probability that it was drawn from bag II.
Problem 25. There are two bags I and II. Bag I
contains 2 white and 4 red balls and Bag II contains 5 white and 3 red balls.
One ball is drawn at random from one of the bags and is found to be red. Find
the probability that it was drawn from bag II
Problem 26. There are two bags I and II. Bag I
contains 3 red and 4 black balls and Bag II contains 5 red and 6 black balls.
One ball is drawn at random from one of the bags and is found to be red. Find
the probability that it was drawn from bag II.
Problem 27. Given three identical boxes I, II
and III, each containing two coins. In box I, both coins are gold coins, in box
II, both are silver coins and in box III, there is one gold and one silver
coin. A person chooses a box at random and takes out a coin. If the coin is of
gold, what is the probability that the other coin in the box is also gold?
Problem 28. A doctor is to visit a patient.
From the past experience, it is known that the probabilities that he will come
by train, bus, and scooter or by other means of transport are respectively. The probability
that he will be late are if he comes by
train bus and scooter respectively. If he comes by other means of transport,
then he will not be late. On a certain day, he arrived late. What is the
probability that he came by train?
Problem 29. Suppose the reliability of a HIV test is
specified as follows: Of people having HIV, 90% of the test detect the disease
but 10% goes undetected (false negative). Of people free of HIV, 99% of the
test are judged HIV negative but 1% are diagnosed as showing HIV positive
(false positive). From a large population of which only 0.1% have HIV, one
person is selected at random, given the HIV test, and the pathologist reports
him/her as HIV positive. What is the probability that the person actually has
HIV?
Problem 30. If a machine is correctly set up, it produces
90% acceptable items. If it is incorrectly set up, it produces only 40%
acceptable items. Past experience shows that 80% of the setups are correctly
done. If after a certain set up, the machine produces 2 acceptable items , find
the probability that the machine is correctly set up.
Problem 31. Assume
that the chances of a patient having a heart attack is 40%. It is also assumed
that a meditation and yoga course reduce the risk of heart attack by 30% and
prescription of certain drug reduces its chances by 25%. At a time a
patient can choose any one of the two options with equal probabilities.
It is given that after going through one of the two options the patient
selected at random suffers a heart attack. Find the probability that the
patient followed a course of meditation and yoga?
Problem 32. Vartika
has an alarm which will ring at the appointed time with probability 0.9. If the
alarm rings, it will awake her and she will reach the examination hall in time
with probability 0.8. If the alarm doesn't ring , she will get up on her own to
reach the examination hall in time, with probability 0.3. Knowing that Vartika
reached the hall in time, find the probability that the alarm rang.
Problem 33. By
examining the chest X−ray probability that T.B is detected when
a person is actually suffering is 0.99. The probability that the doctor
diagnoses incorrectly that a person has T.B on the bases
of X−ray is 0.001. In a certain city 1 in 1000 person
suffers from T.B. A person is selected at random is diagnoses to
have T.B. What is the chance that the actually has T.B?
Problem 34. Shoes are produced by two machines A and B. 50%
of the shoes are produced by machine A with an estimate of 10% of them being
defective. On machine B, 20% of the shoes produced are defective, if a shoe
taken at random is found to be defective , what is the probability that shoe
was produced by machine A?
Problem 35. In a large company, 15% of the employees are
graduates, and of these, 80% work in administrative posts. Of the non-graduates
employees of the company, 10% work in administrative posts. Find the
probability that an employee of this company selected at random from those
working in administrative posts will be graduate.
Problem 36. A pack or cards is counted with face downwards.
It is found that one card is missing. One card is drawn and is found to be red.
Find the probability that the missing card is red.
Problem 37. A card from a pack of 52 cards is lost. From the
remaining cards of the pack, two cards are drawn and are found to be both
spades. Find the probability of the lost card being a spade.
Problem 38. For A,B and C the chances of being selected as
the manager of a firm are in the ratio 4:1:2 respectively. The respective
probabilities for them to introduce a radical change in marketing strategy are
0.3, 0.8 and 0.5. If the change does take place, find the probability that it is
due to the appointment of B or C.
Problem 39. Two
urns I and II contain respectively 3 white and 2 black balls, 2 white and 4
black balls. One ball is transferred from urn I to urn II and then one is drawn
from the latter. It happens to be white. What is the probability that the
transferred ball was white.
Problem 40. A
bin contains 3 different types of disposable flashlights. The probability that
a type 1 flashlight will give over 100 hours of use is 0.7, with the
corresponding probabilities for type 2 and type 3 flashlights being 0.4 and 0.3
respectively. Suppose that 20% of the flashlights in the bin are type 1, 30%
are type 2, and 50% are type 3.
(a) What is the probability that a
randomly chosen flashlight will give more than 100 hours of use?
(b) Given the flashlight lasted over 100
hours, what is the conditional probability that it was a type 2 flashlight?
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