Hello Students! In this post, you will find the complete** ****NCERT Solutions for Maths Class 12 Exercise 13.3**.

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**NCERT Solutions for Maths Class 12 Exercise 13.1**

**NCERT Solutions for Maths Class 12 Exercise 13.2**

**NCERT Solutions for Maths Class
12 Exercise 13.3**

**Maths Class
12 Ex 13.3 Question 1. **

An
urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is
noted and is returned to the urn. Moreover, 2 additional balls of the colour
drawn are put in the urn and then a ball is drawn at random. What is the
probability that the second ball is red?

**Solution:**

The
urn contains 5 red and 5 black balls.

(i) Let a red ball is drawn.

Probability of drawing a red ball = 5/10 = ½

Now, two red balls are added to the urn.

Then, the urn contains 7 red and 5 black balls.

Now, the probability of drawing a red ball = 7/12

(ii) Let a black ball is drawn at first attempt.

Probability of drawing a black ball = 5/10 = ½

Next two black balls are added to the urn.

Now, urn contains 5 red and 7 black balls.

Probability of getting a red ball = 5/12

Then, the probability of drawing a second ball as red

= ½ × 7/12 + ½ × 5/12

= 7/24 + 5/24 = 12/24 = ½

**Maths Class
12 Ex 13.3 Question 2. **

A
bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black
balls. One of the two bags is selected at random and a ball is drawn from the
bag which is found to be red. Find the probability that the ball is drawn from
the first bag.

**Solution:**

Let
A be the event that ball drawn is red and let E1 and E2 be the events that the
ball drawn is from the first bag and second bag respectively.

P(E1)
= ½, P(E2) = ½.

P(A|E1) = Probability of drawing a red ball from bag

**Maths Class
12 Ex 13.3 Question 3. **

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 annual examination. At the end of the year, one student is
chosen at random from the college and he has an A grade, what is the
probability that the student is a hostler?

**Solution:**

Let
E1, E2 and A represents the following:

E1 = students residing in the hostel

E2 = day scholars (not residing in the hostel)

and A = students who attain grade A

**Maths Class
12 Ex 13.3 Question 4. **

In
answering a question on a multiple choice test, a student either knows the
answer or guesses. Let 3/4 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?

**Solution:**

Let
the event E1 = student knows the answer, E2 = he guesses the answer

P(E1) = 3/4, P(E2) = ¼

Let A be the event that answer is correct, if the student knows the answer.

The
answer is correct. Therefore, P(A/E1) = 1

If
he guesses the answer, therefore, P(A/E2) = ¼

Thus,
the probability that a student knows the answer, given that answer is correct,
is

**Maths Class
12 Ex 13.3 Question 5. **

A
laboratory blood test is 99% effective in detecting a certain disease 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 percent
of the population actually has the disease, what is the probability that a
person has the disease given that his test result is positive?

**Solution:**

Let
E1 = The person selected is suffering from certain disease,

E2 = The person selected is not suffering from certain disease

A =
The doctor diagnoses correctly

**Maths Class
12 Ex 13.3 Question 6. **

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 head, what is the probability that it
was the two headed coin?

**Solution:**

Let
E1, E2, E3 and A denote the following:

E1 = a two headed coin, E2 = a biased coin,

E3 = an unbiased coin, A = A head is shown

**Maths Class
12 Ex 13.3 Question 7. **

An
insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck
drivers. The probability of accidents are 0.01, 0.03, 0.15 respectively. One of
the insured persons meets with an accident. What is the probability that he is
a scooter driver?

**Solution:**

Let
E1 = the person selected is a scooter driver, E2 = the person selected is a car
driver, E3 = the person selected is a truck driver and A = the person meets
with an accident

Total
number of drivers = 2000 + 4000 + 6000 = 12,000

Probability of selecting a scooter driver

**Maths Class
12 Ex 13.3 Question 8. **

A
factory has two machines A and B. Past record shows that machine A produced 60%
of the items of output and machine B produced 40% of the items. Further, 2% of
the items produced by machine A and 1% produced by machine B were defective. All
the items are put into one stockpile and then one item is chosen at random from
this and is found to be defective. What is the probability that it was produced
by machine B?

**Solution:**

Let
E_{1} and E_{2} be the events the percentage of
production of items by machine A and machine B respectively.

Let A denotes defective item.

Machine A’s production of items = 60%

Probability of production of items by machine A

**Maths Class
12 Ex 13.3 Question 9. **

Two
groups are competing for the position on the Board of directors of a
corporation. The probabilities that the first and the second groups 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.

**Solution:**

Given:
P(G1) = 0.6, P(G2) = 0.4

Let P represents the launching of new product, then P(P|G1) = 0.7 and P(P|G2) =
0.3

**Maths Class
12 Ex 13.3 Question 10. **

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?

**Solution:**

**Maths Class
12 Ex 13.3 Question 11.**

A
manufacturer has three machine operators A, B and C. The first operator A produces
1% defective items, whereas 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 the
job for 30% of the time and C is on the job for 20% of the time. A defective
item is produced, what is the probability that it was produced by A?

**Solution:**

Let
E1, E2, E3 and A be the events defined as follows:

E1 = the item is manufactured by the operator A

E2 = the item is manufactured by the operator B

E3 = the item is manufactured by the operator C

and A = the item is defective

**Maths Class
12 Ex 13.3 Question 12. **

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?

**Solution:**

Let
E1 = Event that lost card is diamond,

E2 = Event that lost card is not diamond.

There are 13 diamond cards, out of a pack or 52 cards.

**Maths Class
12 Ex 13.3 Question 13. **

Probability
that A speaks truth is 4/5. A coin is tossed. A reports that a head appears.
The probability that actually there was head is:

(A) 4/5

(B) ½

(C) 1/5

(D) 2/5

**Solution:**

(A)
Let A be the event that the man reports that head occurs in tossing a coin and
let E1 be the event that head occurs and E2 be the event head does not occurs.

**Maths Class
12 Ex 13.3 Question 14. **

If
A and B are two events such that A ⊂ B and P(B) ≠ 0, then
which of the following is correct:

(A) P(A|B) = P(B)/P(A)

(B) P(A|B) < P(A)

(C) P(A|B) ≥ P(A)

(D) None of these

**Solution:**

(C)
A ⊂ B implies that A∩B = A and P(B) ≠ 0

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