UPSC NDA Math Model Paper 1

Given, a man speaks truth 2 out of 3 times. He picks one of the natural numbers in the set S = {1, 2, 3, 4, 5, 6, 7} and reports that it is even.
E1= the event of an even number is picked
E2= the event of an odd number is picked
⇒P(E1)=

3

7

⇒P(E2)=

4

7

E= the event a man reports that it is even
A man speaks truth 2 out of 3 times.
⇒P(E|E1)=

2

3

⇒P(E|E2)=

1

3

To find: The probability that it is actually even i.e. P(E1|E)
P(E1|E) =

P(E|E1).P(E1)

P(E|E1).P(E1)+P(E|E2).P(E2)

⇒P(E1|E) =

( 2 3 ).( 5 7 )

( 2 3 ).( 3 7 )+( 1 3 ).( 4 7 )

⇒P(E1|E)=

6

10

⇒P(E1|E)=

3

5

Hence, a man speaks truth 2 out of 3 times. He picks one of the natural numbers in the set S = {1, 2, 3, 4, 5, 6, 7} and reports that it is even. The probability that it is actually even is

3

5