Question 1206432
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.