Question

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?

Answer 1

Let, P( B ) and P( R ) be the probabilities.

The probability of first ball drawn black is P( B ).

The probability of second ball drawn Red is P( R ).

Now, the probability that the second ball drawn is red given by,

P( R )=( P( B )×P( R B ) )+( P( B ′ )×P( R B ′ ) )(1)

Here,

Probability of second ball is red,

P( B )= ( Total Red balls in the urn ) ( total red balls in the urn )+( total black balls in the urn ) = 5 5+5 = 1 2

P( B ′ )=1−P( B ) = 1 2

Now, there are 5 red and ( 5+2 ) black balls in the urn.

P( R B )= 5 5+7 = 5 12 P( R ′ B )=1−P( R B ) = 7 12

Put these values in equation (1).

P( R )=( P( B )×P( R B ) )+( P( B ′ )×P( R B ′ ) ) =( 1 2 × 5 12 )+( 1 2 × 7 12 ) = 12 24 = 1 2

Thus, the probability that the second ball is red is 1 2 .