Question

Suppose that there are 2 red balls, 1 blue ball, and 2 white balls in an urn. A ball is drawn and its color is noted. After the ball is drawn, it is set aside and is replaced with a blue ball. Another ball is then drawn from the urn. Find the probability the first ball drawn was red given that the second ball drawn was blue.

Answer 1

The total number of balls remains at 5 for each draw as we replace the ball with a blue ball after the first draw,

the probability that the first ball drawn is red is

2 P(first red) = number of red balls total number of balls5

Having drawn the first ball as red, we replace the red ball with a blue ball. The urn now has 1 red, 2 blue and 2 white balls.

the probability that the second ball drawn is blue given that the first is red is

P(second blue first red) = number of blue balls total number of balls 2 5

the probability that the first is red and the second ball drawn is blue is

P(first red n second blue) = P(second blue first red) P(first red) using the formula for conditional probability GIN

the probability that the first ball drawn is blue is

number of blue balls P(first blue) = 1 total number of balls 1 5

Having drawn the first ball as blue, we replace the blue ball with a blue ball. The urn now has 2 red, 1 blue and 2 white balls.

the probability that the second ball drawn is blue given that the first is blue is

1 P(second blue first blue) = number of blue balls total number of balls5

the probability that the first is blue and the second ball drawn is blue is

P(first blue second blue) = P(second blue first blue) P(first blue) using the formula for conditional probability - X

the probability that the first ball drawn is white is

P(first white) = number of white balls total number of balls 5

Having drawn the first ball as white, we replace the white ball with a blue ball. The urn now has 2 red, 2 blue and 1 white balls.

the probability that the second ball drawn is blue given that the first is white is

number of blue balls P(second blue first white) = total number of balls 2 5

the probability that the first is white and the second ball drawn is blue is

P(first white n second blue) = P(second blue first whiteP(first white) using the formula for conditional probability

Now the probability that is needed

the probability the first ball drawn was red given that the second ball drawn was blue is

P(first red n second blue) P(first red second blue) = - 2 using the formula for conditional probability P(second blue) using the formula for total probability we get P(first red n second blue) P(first red n second blue) + P(first blue second blue) + P(first whiten second blue) + +

ans:  the probability the first ball drawn was red given that the second ball drawn was blue is 4/9=0.4444