Ball Bag
A bag contains one white ball and two red balls. A ball is drawn at random. If the ball is white it is put b
A bag contains one white ball and two red balls. A ball is drawn at random. If the ball is white it is put back in the bag along with another white ball. If the ball is red it is put back in the bag along with two extra red balls.
a) Find the probability that the second ball drawn is red.
b) If the second ball drawn was red what is the probability that the first ball drawn was red?
Let A be the event that the first ball is white, A’ first is red
B second is red; B’ second is white.
If A, then the second draw is from 2 white and 2 red, and so
P(B|A) = 2/4 = 1/2;
and of course P(B’|A) = 1/2
where P(B|A) means the probability of B given A has occurred.
If B, there are 4 red and 1 white for the second draw, and so
P(B|A’) = 1/5 and P(B’|A’) = 4/5
The second ball is red if A occurs followed by B’, or A’ occurs followed by B’.
Now P(AB’) = P(A)*P(B’|A)
………………. = 1/3 * 1/2
……………….. = 1/6
P(A’B') = P(A’)*P(B’|A’)
………….= 2/3 * 4/5
………… = 8/15
Therefore P(B’) = 1/6 + 8/15
……………………. = 5/30 + 16/30
……………………. = 21/30
……………………. = 7/10
b) From above, if we look at 30 equally likely outcomes, B’ occurs in 21 of these outcomes.
Of these 21, there are 5 where A occurs and 16 where A’ [i.e. first ball red] occurs.
Therefore prob that first was red = 16/21.
Formally,
P(A’B') = P(A’|B’) * P(B’)
We have already found that
P(A’B') = 8/15 and P(B’) = 7/10
Therefore
8/15 = P(A’|B’) * 7/10
P(A’|B’) = 8/15 * 10/7 = 16/21
Frankie’s Ball Bag
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