Solved problems on theorem of total probability for CBSE, ISC and other state boards in India.
Problem 1. A bag contains 3 white and 2 red balls and
another bag contains 2 white and 4 red balls. A ball is taken from the first
bag and without seeing its colour is put in the second bag. A ball is taken
from the latter. Find the probability that the ball drawn is red.
Problem 2. The ratio of the number of boys to the number of
girls in a class is 1:2. It is known that the probabilities of a girl and a boy
getting a first division are 0.25 and 0.28 respectively. Find the probability
that a student chosen at random will get first division.
Problem 3. A bag contains 3 white and 6 black balls and
another bag contains 6 white and 3 black balls. A bag is selected at random and
a ball is drawn. Find the probability that the ball drawn is of white colour.
Problem 4. Two machines A and B produce respectively 60% and
40% of the total number of items of a factory. The percentage of defective
output of these machines are respectively 2% and 5%. If an item is selected at
random, what is the probability that the item is defective?
Problem 5. There are two bags. The first bag contains 5 red
and 3 black balls and the second bag contains 3 red and 5 black balls. Two
balls are drawn at random from the first bag and are put into the second bag
without noting their colours. Then two balls are drawn from the second bag.
Find the probability that the balls drawn are one red and one black.
Problem 6. There are two bags. The first bag contains 5 red
and 3 black balls and the second bag contains 3 red and 5 black balls. Two
balls are drawn at random from the first bag and are put into the second bag
without noting their colours. Then two balls are drawn from the second bag.
Find the probability that the balls drawn are one red and one black.
Problem 7. A bag X contains 3 white and 2 black balls and
another bag Y contains 2 white and 4 black balls. A bag and a ball out of it is
picked at random. What is the probability that the ball is white?
Problem 8. An urn contains m white and n black balls. A ball
is drawn at random and is put back into the urn along with k additional balls
of the same colour as that of the ball drawn. A ball is again drawn at random.
What is the probability that the ball drawn now is white?
Problem 9. An urn contains m white and n black balls. A ball
is drawn at random and is put back into the urn along with k additional balls
of the same colour as that of the ball drawn. A ball is again drawn at random.
What is the probability that the ball drawn now is white?
Problem 10. A person has undertaken a construction job. The
probabilities are 0.65 that there will be strike, 0.80 that the construction
job will be completed on time if there is no strike, and 0.32 that the
construction job will be completed in time if there is a strike. Determine the
probability that the construction job will be completed in time.
Problem 11. A survey is carried out to find the percentage
of men who sing in the bathroom. Because some people may be too embarrassed to
admit openly to be bathroom singers, each person question is asked to roll a
die in secret and answer NO if the number shown is 1 and YES if it is 6, no
matter what the true answer is, but to tell the truth (YES or NO) if 2,3,4 or 5
comes out. Because the number shown is not revealed, it is impossible to tell
from the answer given whether the person is a bathroom singer or not. Suppose
that the probability of answering YES in this survey is found to be 2/3. What
is the probability of being a bathroom singer, then?
