Question 1206432: An urn contains 5 red and 7 blue balls. Suppose that two balls are selected randomly and with replacement. Let A and B be the events that the first and the second balls are red respectively.
(a)Is event A and B independent？Show your reasoning.
(b)If we do the same experiment without replacement, is event A and event B independent? Show your reasoning.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! independence means the probability is not affected by what came before.
the urn contains 5 red and 7 blue balls.
you select 2 balls at random, with replacement.
you want A and B the event of drawing a red ball.
the first time you draw a ball randomly, the probability that the ball is red is 5 / 12.
that's because there are 5 red balls in the pot out of a total of 12 balls (5 red plus 7 blue = 12 total).
after you put the ball back in the pot, the probability of drawing a red ball the second time is also 5 / 12.
the probability is the same because of the replacement.
you had 5 red out of 12 the first time and you have 5 red out of 12 the second time.
that makes the 2 draws independent of each other.
if you draw the red ball the first time and don't replace it, then the probability of drawing a red ball the second time changes.
it is not 4 / 11.
that's because there are now 4 red balls in the pot out of a total of 11.
the probability of drawing a red ball the second time is different because it was affected by what you draw the second time.
that makes the probability of drawing a red ball the second time dependent on what happened the first time.
note that the probability of drawing a red ball the second time is changed whether or not you drew a red ball the first time.
if you drew a red ball the first time, the probability of drawing a red ball the second time is 4/11.
if you drew a blue ball the first time, the probability of drawing a red ball the second time is 5/11.
you still have 5 red balls in the pot, but the total number of balls in the pot changed from 12 to 11.
bottom line:
with replacement, probability of drawing a red ball on the second draw is the same as the probability of drawing a red ball on the first draw because it was not affected by what happened on the first draw.
without replacement, probability of drawing a red ball on the second draw is different from the probability of drawing a red ball on the first draw because it was affected by what happened on the first draw.
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