An urn contains 10 white and 3 black balls another urn contains 3 white and 5 black balls - Maths - Probability

Question

An urn contains 10 white and 3 black balls another urn contains
3 white and 5 black balls Two are drawn from the first urn and put
into the second urn and then a ball is drawn from the latter Find
the probability that it is a white ball

Aakash Sharma · Accepted Answer

A white all can be drawn from second urn in three mutually exclusive ways

(1) By transfering two white balls from first urn to second urn and then drawing white ball from it.

(2) By transfering two black balls from first urn to second urn and then drawing white ball from it.

(3) By transfering one black and one white ball from first urn to second urn and then drawing a white ball from it.

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

E₁= Two white balls are transferred from the first urn to the second urn

E₂= Two black balls are transferred from the first urn to the second urn

E₃ = One black and one white ball are transferred from the first urn to the second

A = a white ball is drawn from the second urn

Since first urn contains 10 white and 3 black balls, we have

$P (E_{1}) = \frac{10_{C_{2}}}{13_{C_{2}}} = \frac{45}{78} P (E_{2}) = \frac{3_{C_{2}}}{13_{C_{2}}} = \frac{3}{78} P (E_{3}) = \frac{10_{C_{1}} \times 3_{C_{1}}}{1 3_{C_{2}}} = \frac{30}{78}$

If E₁ has already occurred then second urn containing 5 white and 5 black balls so,

$P (\frac{A}{E_{1}}) = \frac{5}{10}$

If E₂ has already occurred then second urn contains 3 white and 7 black balls so,

$P (\frac{A}{E_{2}}) = \frac{3}{10}$

If E₃ has already occurred then the second urn contains 4 white and 6 black balls so,

$P (\frac{A}{E_{3}}) = \frac{4}{10}$

Hence by law of probability

$P (Getting a white ball) = P (A) = P (E_{1}) P (\frac{A}{E_{1}}) + P (E_{2}) P (\frac{A}{E_{2}}) + P (E_{3}) P (\frac{A}{E_{3}}) = \frac{45}{78} \times \frac{5}{10} + \frac{3}{78} \times \frac{3}{10} + \frac{30}{78} \times \frac{4}{10} = \frac{354}{780} = \frac{59}{130}$

An urn contains 10 white and 3 black balls. another urn contains 3 white and 5 black balls. Two are drawn from the first urn and put into the second urn and then a ball is drawn from the latter. Find the probability that it is a white ball.