Question

Two balls are drawn in succession out of a box containing 2 red and 5 white balls. Find the probability that at least 1 ball was​ red, given that the first ball was

(A) Replaced before the second draw.
(B) Not replaced before the second draw.

(A) Find the probability that at least 1 ball was​ red, given that the first ball was replaced before the second draw.

​(Simplify your answer. Type an integer or a​ fraction.)

​(B) Find the probability that at least 1 ball was​ red, given that the first ball was not replaced before the second draw.

​(Simplify your answer. Type an integer or a​ fraction.)

Answer 1

a) first ball replaced before the second is drawn:

to draw at least 1 red ball = 1 - probability of two whites

first time to draw a white = 5/7

when drawing the second time, you have the same probability to get a white as the first time since the first ball is put back

therefore probability of at least 1 red with replacement = 1 - (5/7 * 5/7) = 0.489796

b) without replacement:

the first time you get a white, probability is 5/7

if you got a white, now you have three-four balls left and a total of 6 balls,

so the second time you get a white will have a probability of 4/6

therefore the probability of getting two whites without replacement = 5/7 * 4/6

to draw at least 1 red ball = 1 - probability of two whites = 1 - 5/7 * 4/6 = 0.52381