Probability

Bag P contains 3 white, 2 red, and 5 blue balls, while bag Q contains 2 white, 3 red, and 5 blue balls. We are interested in finding the probability of drawing a red ball from bag Q after transferring one ball from bag P to bag Q.
First, define E1,E2, and E3 as the events of drawing a white, red, and blue ball from bag P, respectively. The probabilities are calculated as follows:
P(E1)=‌

3

10

(probability of drawing a white ball from P )
P(E2)=‌

2

10

(probability of drawing a red ball from P )
P(E3)=‌

5

10

(probability of drawing a blue ball from P )
Now, consider the composition of bag Q after transferring one ball from bag P :
If E1 occurs (a white ball is transferred), bag Q will have 3 white balls, 3 red balls, and 5 blue balls. The probability of drawing a red ball from Q becomes ‌

3

11

.
If E2 occurs (a red ball is transferred), bag Q will have 2 white balls, 4 red balls, and 5 blue balls. The probability of drawing a red ball from Q becomes ‌

4

11

.
If E3 occurs (a blue ball is transferred), bag Q will have 2 white balls, 3 red balls, and 6 blue balls. The probability of drawing a red ball from Q becomes ‌

3

11

.

The overall probability P(E) of drawing a red ball from bag Q is computed using the law of total probability:
P(E)=P(E1)⋅‌

3

11

+P(E2)⋅‌

4

11

+P(E3)⋅‌

3

11

Substitute the values:
‌P(E)=(‌

3

10

×‌

3

11

)+(‌

2

10

×‌

4

11

)+(‌

5

10

×‌

3

11

)
‌P(E)=‌

9

110

+‌

8

110

+‌

15

110

=‌

32

110

=‌

16

55

The probability that a ball chosen from bag Q is red after transfer is ‌

16

55

.