Bag A contains 3 red balls and 4 white balls. Bag B contains 2 red balls and 3 white balls. A ball is transferred from A to B, and then a ball is drawn from bag B. What is the probability that the ball thus drawn will be white?

Asked by bjayanta | 9th Apr, 2017, 11:21: AM

Expert Answer:

begin mathsize 14px style straight A space white space ball space can space be space drawn space from space the space second space bag space in space two space mutually space exclusive space ways colon
1 right parenthesis space By space transfering space straight a space white space ball space from space first space bag space to space the space second space bag space and space then space drawing space straight a space white space ball space from space it.
2 right parenthesis space By space transfering space straight a space red space ball space from space first space bag space to space the space second space bag space and space then space drawing space straight a space white space ball space from space it.

Let space straight E subscript 1 comma space straight E subscript 2 space and space straight A space be space the space events space defined space as space follows colon
straight E subscript 1 equals straight a space white space ball space is space transferred space from space the space first space bag space to space the space second space bag
straight E subscript 2 equals straight a space red space ball space is space transferred space from space the space first space bag space to space the space second space bag
straight A equals straight a space white space ball space is space drawn space from space the space second space bag

Since space the space first space bag space contains space 3 space red space and space 4 space white space balls comma space we space have
straight P left parenthesis straight E subscript 1 right parenthesis equals 4 over 7 space and space straight P left parenthesis straight E subscript 2 right parenthesis equals 3 over 7
If space straight E subscript 1 space has space already space occured comma space that space is comma space straight a space white space ball space has space already space been space transferred space from space first space bag space to space the space second
bag comma space then space the space second space bag space contains space 2 space red space and space 4 space white space balls.
So comma space straight P left parenthesis straight A divided by straight E subscript 1 right parenthesis equals 4 over 6
If space straight E subscript 2 space has space already space occured comma space that space is comma space straight a space red space ball space has space been space transferred space from space first space bag space to space the space second
bag comma space then space the space second space bag space contains space 3 space red space and space 3 space white space balls.
So comma space straight P left parenthesis straight A divided by straight E subscript 2 right parenthesis equals 3 over 6
By space the space law space of space total space probability comma space we space have
straight P left parenthesis getting space straight a space white space ball right parenthesis
equals straight P left parenthesis straight A right parenthesis
equals straight P left parenthesis straight E subscript 1 right parenthesis straight P left parenthesis straight A divided by straight E subscript 1 right parenthesis plus straight P left parenthesis straight E subscript 2 right parenthesis straight P left parenthesis straight A divided by straight E subscript 2 right parenthesis
equals 4 over 7 cross times 4 over 6 plus 3 over 7 cross times 3 over 6
equals 16 over 42 plus 9 over 42
equals 25 over 42 end style

Answered by Rashmi Khot | 10th Apr, 2017, 10:21: AM