UPSC NDA I 2025 Mathematics Paper

Given,
The probability that A,B, and C become managers are:
P(A)=‌

3

10

,P(B)=‌

1

2

,P(C)=‌

4

5

The conditional probabilities that the bonus scheme is introduced are:
P(D∣A)=‌

4

9

,P(D∣B)=‌

2

9

,P(D∣C)=‌

1

3

We need to find the probability that B is the manager given that the bonus scheme has been introduced using Bayes' Theorem:
P(B∣D)=‌

P(D∣B)P(B)

P(D)

First, calculate the total probability P(D) that the bonus scheme is introduced:
P(D)=P(D∣A)P(A)+P(D∣B)P(B)+P(D∣C)P(C)
Substitute the values:
P(D)=(‌

4

9

×‌

3

10

)+(‌

2

9

×‌

1

2

)+(‌

1

3

×‌

4

5

)
Simplifying:
P(D)=‌

12

90

+‌

2

18

+‌

4

15

Find a common denominator ( LCD=90 ):
P(D)=‌

12

90

+‌

10

90

+‌

24

90

=‌

46

90

=‌

23

45

Now, use Bayes' Theorem to find P(BID) :
P(B∣D)=‌

P(D∣B)P(B)

P(D)

=‌

( 2 9 × 1 2 )

23 45

Simplifying:
P(B∣D)=‌

‌ 2 18

23 45

=‌

‌ 1 9

23 45

=‌

5

23

The probability that the manager appointed was B, given that the bonus scheme has been introduced, is 5∕23.