Box A contains six white marbles and five black marbles. Box B contains three white marbles and four black marbles. An experiment consists of first selecting a marble at random from Box A. The marble is transferred to Box B and then a second marble is drawn from Box B. What is the probability that the first marble was white given that the second marble was white?
We have four possible outcomes:
1 ww **
2 wb
3 bw
4 bb
we want the cases marked by **
#1:
ww, prob(first w) = 6/11
so the white gets transferred to B and
prob(2nd w) = 4/8 = 1/2
prob(ww) = (6/11)(1/2) = 6/22 = 3/11
well,
P(BW) = 5/11 * 3/8
P(WW) = 6/11 * 4/8
so, what does hat tell you?
To find the probability that the first marble was white given that the second marble was white, we can use Bayes' theorem.
Let's denote the events as follows:
A1: First marble is white
A2: Second marble is white
We're interested in finding P(A1|A2), which is the probability that the first marble was white, given that the second marble was white.
According to Bayes' theorem:
P(A1|A2) = (P(A2|A1) * P(A1)) / P(A2)
P(A2|A1) represents the probability of drawing a white marble from Box B, given that the first marble was white. Since the first marble is selected from Box A, this probability can be found by taking the number of white marbles in Box B divided by the total number of marbles transferred from Box A to Box B.
P(A1) represents the probability of drawing a white marble from Box A initially. This can be found by taking the number of white marbles in Box A divided by the total number of marbles in Box A.
P(A2) represents the probability of drawing a white marble from Box B without considering any initial information. This can be found by taking the number of white marbles in Box B before the transfer divided by the total number of marbles in Box B before the transfer.
Let's calculate each of these probabilities step-by-step.
Step 1: Calculate P(A2|A1)
Since one white marble is transferred from Box A to Box B, the probability of drawing a white marble from Box B, given that the first marble was white, can be found as follows:
P(A2|A1) = (number of white marbles in Box B) / (total number of marbles transferred from Box A to Box B)
= 3 / 11
Step 2: Calculate P(A1)
The probability of drawing a white marble from Box A initially can be calculated as follows:
P(A1) = (number of white marbles in Box A) / (total number of marbles in Box A)
= 6 / 11
Step 3: Calculate P(A2)
The probability of drawing a white marble from Box B without considering any initial information can be calculated as follows:
P(A2) = (number of white marbles in Box B before transfer) / (total number of marbles in Box B before transfer)
= (3 + 1) / (6 + 5)
= 4 / 11
Step 4: Calculate P(A1|A2)
Finally, we can use Bayes' theorem to find the probability of the first marble being white, given that the second marble was white:
P(A1|A2) = (P(A2|A1) * P(A1)) / P(A2)
= [(3/11) * (6/11)] / (4/11)
= 18/44
= 9/22
Therefore, the probability that the first marble was white, given that the second marble was white, is 9/22 or approximately 0.409.
To find the probability that the first marble drawn was white given that the second marble was white, we can use conditional probability. We need to find the probability of both marbles being white and then divide it by the probability of the second marble being white.
Step 1: Calculate the probability of both marbles being white.
The probability of drawing a white marble from Box A is 6/11 (6 white out of 11 total marbles).
After transferring the marble from Box A to Box B, the total number of marbles in Box B becomes 10 (since 1 marble was transferred).
The probability of drawing a white marble from Box B, given that the marble transferred from Box A was white, is (7 white marbles out of 10 total).
To find the probability of both marbles being white, we multiply the probabilities:
(6/11) * (7/10) = 42/110 = 21/55
Step 2: Calculate the probability of the second marble being white.
The probability of drawing a white marble from Box B is 7/10 (7 white out of 10 total marbles).
Step 3: Calculate the conditional probability.
To find the probability that the first marble was white given that the second marble was white, we divide the probability of both marbles being white (from Step 1) by the probability of the second marble being white (from Step 2):
(21/55) / (7/10) = (21/55) * (10/7) = 6/11
Thus, the probability that the first marble drawn was white given that the second marble was white is 6/11.