Probability - Exemplar Solutions

CBSE Class–12 Mathematics

NCERT Exemplar
Chapter 13
Probability - Short Answer Questions

1. A and B are two candidates seeking admission in a college. The probability that A is selected is 0.7 and the probability that exactly one of them is selected is 0.6. Find the probability that B is selected.

Sol. Let p be the probability that B gets selected.

P(A is selected)=0.7, $P (A^{'}) = 1 - P (A) = 1 - 0.7 = 0.3$

P (Exactly one of A, B is selected) = 0.6

P (A is selected, B is not selected;B is selected, A is not selected) = 0.6

Thus, the probability that B gets selected is 0.25.

2. The probability of simultaneous occurrence of at least one of two events A and B is p. If the probability that exactly one of A, B occurs is q, then prove that .

Sol. Since P (exactly one of A, B occurs) = q(given), we get

3. 10% of the bulbs produced in a factory are of red colour and 2% are red and defective. If one bulb is picked up at random, determine the probability of its being defective if it is red.

Sol. Let A and B be the events that the bulb is red and defective respectively.

Thus, the probability of the picked up bulb of its being defective, if it is red, is .

4. Two dice are thrown together. Let A be the event ‘getting 6 on the first die’ and B be the event ‘getting 2 on the second die’. Are the events A and B independent?

Sol. A = {(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

B = {(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2)}

Events A and B will be independent if

i.e.,

Here LHS=RHS

Hence, A and B are independent events.

5. A committee of 4 students is selected at random from a group consisting 8 boys and 4 girls. Given that there is at least one girl in the committee, calculate the probability that there are exactly 2 girls in the committee.

Sol. Let A denote the event that at least one girl will be chosen, and B the event that exactly 2 girls will be chosen. We require .

Since A denotes the event that at least one girl will be chosen, A’ denotes that no girl is chosen, i.e., 4 boys are chosen. Then

$[^{n} C_{r} = \frac{n!}{r! (n - r)!}]$

$P (A) = 1 - \frac{14}{99} = \frac{85}{99}$

Now

Thus

6. Three machines E1, E2, E3in a certain factory produce 50%, 25% and 25%, respectively, of the total daily output of electric tubes. It is known that 4% of the tubes produced one each of machines E1and E2are defective, and that 5% of those produced on E3are defective. If one tube is picked up at random from a day’s production, calculate the probability that it is defective.

Sol. Let D be the event that the picked up tube is defective.

Let be the events that the tube is produced on machinesrespectively.

Also

Putting these values in (1), we get

7. Find the probability that in 10 throws of a fair die a score which is a multiple of 3 will be obtained in at least 8 of the throws.

Sol. Here success is a score which is a multiple of 3 i.e., 3 or 6.

Therefore,

The probability of r success in 10 throws is given by

Now P (at least 8 successes) = P (8) + P (9) + P (10)

8. A discrete random variable X has the following probability distribution:

X	1	2	3	4	5	6	7
P(X)	C	2C	2C	3C	C2	2C2	7C2+C

Find the value of C. Also find the mean of the distribution.

Sol. Since , we have

either (10C- 1)=0 or (C+1)=0

(C=-1 cannot be possible.probability cananot be negative)

Therefore, the permissible value of .

Mean =

Long Answer

9. Four balls are to be drawn without replacement from a box containing 8 red and 4 white balls. If X denotes the number of red ball drawn, find the probability distribution of X.

Sol. Since 4 balls have to be drawn, therefore, X can take the values 0, 1, 2, 3, 4.

= P (no red ball) = P (4 white balls)

= P (1 red ball and 3 white balls)

= P (2 red balls and 2 white balls)

= P (3 red balls and 1 white ball)

= P (4 red balls)

Thus, the following is the required probability distribution of X

10. Determine variance and standard deviation of the number of heads in three tosses of a coin.

Sol. Let X denote the number of heads tossed. So, X can take the values 0, 1, 2, 3. When a coin is tossed three times, we get

Sample space S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

= P (no head) = P (TTT)

= P (one head) = P (HTT, THT, TTH)

= P (two heads) = P (HHT, HTH, THH)

= P (three heads) = P (HHH)

Thus the probability distribution of X is:

Variance of X

where μ is the mean of X given by

Now

From (1), (2) and (3), we get

Standard deviation

11. Refer to Example 6. Calculate the probability that the defective tube was produced on machine E1.

Sol. Now, we have to find .

12. A car manufacturing factory has two plants, X and Y. Plant X manufactures 70% of cars and plant Y manufactures 30%. 80% of the cars at plant X and 90% of the cars at plant Y are rated of standard quality. A car is chosen at random and is found to be of standard quality. What is the probability that it has come from plant X?

Sol. Let E be the event that the car is of standard quality. Let be the events that the car is manufactured in plants X and Y, respectively. Now