Question

Bag A contains 3 red and 2 white balls and bag B contains 2 red and 5 white balls. A bag is selected at random, a ball is drawn and put into the other bag; and then a ball is drawn from the second bag. Find the probability that both the balls drawn are of the same colour.

Answer 1

According to the question,there are two bags,

Let choosing bag A is event A,so probability of choosing bag A, p(A)= 1/2

Let choosing bag B is event A,so probability of choosing bag B, p(B)= 1/2

Case 1:

Let drawing Red ball from bag A,then probability
$p\bigg( \frac{R}{A}\bigg ) = \frac{3}{5}$
and into bag B,Now drawing Red ball from bag B,so Probability of drawing Red ball from bag B
$p\bigg( \frac{R}{B}\bigg) = \frac{3}{8}$
By the same way

Case2:

Let drawing white ball from bag A,then probability
$p \bigg( \frac{W}{A}\bigg) = \frac{2}{5}$

and drop into bag B,Now drawing white ball from bag B,so Probability of drawing white ball from bag B
$p\bigg ( \frac{W}{B}\bigg) = \frac{6}{8}$

Thus , Probability of drawing a ball of same colour is
$p(A)p\bigg( \frac{R}{A}\bigg ) \times p(B)p\bigg( \frac{R}{B}\bigg ) + p(A)p\bigg( \frac{W}{A}\bigg ) \times p(B)p\bigg( \frac{W}{B}\bigg ) \\ \\ = \frac{1}{2} \times \frac{3}{5} \times \frac{1}{2} \times \frac{3}{8} + \frac{1}{2} \times \frac{2}{5} \times \frac{1}{2} \times \frac{6}{8} \\ \\ = \frac{9}{160} + \frac{12}{160} \\ \\ = \frac{21}{160} \\ \\ = 0.131$

Hope it helps you

Answer 2

Answer:

0.536

Step-by-step explanation:

Hi,

Given that Bag A contains 3 Red and 2 White balls,

Bag B contains 2 red and 5 white balls

Let 'Fra' be the event that the first ball drawn is Red  and is from

Bag A

Let 'Fwa' be the event that the first ball drawn is White and is

from Bag A

Let 'Frb' be the event that the first ball drawn is Red and is from

Bag B

Let 'Fwb' be the event that the first ball drawn is White and is

from Bag B

Let 'Sra' be the event that the second ball drawn is Red and is

from Bag A

Let 'Swa' be the event that the second ball drawn is White and is

from Bag A

Let 'Srb' be the event that the second ball drawn is Red and is

from Bag B

Let 'Swb' be the event that the second ball drawn is White and is

from Bag B,

Since a bag is selected at random and any of the bags A or B

could be chosen at first place,

Let 'Ba' be the event of choosing Bag A at first place

P(Ba) = 1/2

Let 'Bb' be the event of choosing Bag B at first place

P(Bb) = 1/2

Probability that both the balls drawn are of same colour

= P(Ba)(Probability that both balls drawn are of same colour /Ba)

+ P(Bb)(Probability that both balls drawn are of same colour/Bb)

So, (Probability that both balls drawn are of same colour /Ba)

= P(Fra)P(Srb/Fra) + P(Fwa)P(Swb/Fwa)

= 3/5*3/8 + 2/5*6/8

= 21/40

(Probability that both balls drawn are of same colour /Ba)

= P(Frb)P(Sra/Frb) + P(Fwb)P(Swa/Fwb)

= 2/7*4/6 + 5/7*3/6

=  23/42

Probability that both the balls drawn are of same colour

= 1/2*21/40 + 1/2*23/42

= 3604/6720

= 901/1680

= 0.536

Hope, it helps !