Box A contains 3 white marbles and 4 black marbles. Box B contains 6 white marbles and 5 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?
To find the probability that the first marble was white given that the second marble was white, we need to use conditional probability.
Let's break down the problem step-by-step:
Step 1: Determine the probability of getting a white marble as the first pick from Box A.
The total number of marbles in Box A is 7 (3 white + 4 black).
Therefore, the probability of selecting a white marble from Box A on the first pick is 3/7.
Step 2: Determine the probability of getting a white marble as the second pick from Box B, given that the first marble was white.
After transferring the marble from Box A to Box B, the total number of marbles in Box B becomes 13 (6 white + 7 black).
Since the first marble was white, we now have 6 white marbles out of 13 in Box B.
Therefore, the probability of selecting a white marble from Box B on the second pick, given that the first marble was white, is 6/13.
Step 3: Calculate the conditional probability.
To calculate the conditional probability, we divide the probability of both events happening (getting a white marble on the first pick and getting a white marble on the second pick) by the probability of the second event occurring (getting a white marble on the second pick).
Conditional Probability = P(First marble is white and second marble is white) / P(Second marble is white)
The probability of both events happening (getting a white marble on the first pick and getting a white marble on the second pick) is (3/7) * (6/13).
The probability of the second event occurring (getting a white marble on the second pick) is 6/13.
Therefore, the conditional probability is:
(3/7) * (6/13) / (6/13) = 3/7
So, the probability that the first marble was white given that the second marble was white is 3/7.
To find the probability that the first marble was white given that the second marble was white, we can use Bayes' theorem. Bayes' theorem states that:
P(A|B) = (P(B|A) * P(A)) / P(B)
Where:
P(A|B) is the probability of A given B.
P(B|A) is the probability of B given A.
P(A) is the probability of A.
P(B) is the probability of B.
In this case, we want to find P(First marble is white | Second marble is white). Let's break down the problem:
1. Calculate the probability of the second marble being white:
P(Second marble is white) = P(Second marble is white | First marble is white) * P(First marble is white) + P(Second marble is white | First marble is black) * P(First marble is black)
In this case, there are two possibilities for the first marble: white or black. We need to calculate the probability of drawing a white marble from Box B under these two scenarios.
- If the first marble is white, the probability of drawing the second marble white is:
P(Second marble is white | First marble is white) = (6/12) = 1/2 (Because there are 6 white marbles and 12 total marbles in Box B after one marble is transferred from Box A)
- If the first marble is black, the probability of drawing the second marble white is:
P(Second marble is white | First marble is black) = (5/12) (Because there are 5 white marbles and 12 total marbles in Box B after one marble is transferred from Box A)
The probability of the first marble being white is given as P(White) = (3/7) (Because there are 3 white marbles and 7 total marbles in Box A).
So, substituting these values into the equation:
P(Second marble is white) = (1/2) * (3/7) + (5/12) * (4/7)
2. Calculate the probability of the first marble being white given that the second marble is white:
P(First marble is white | Second marble is white) = (P(Second marble is white | First marble is white) * P(First marble is white)) / P(Second marble is white)
Again, substituting the values we found:
P(First marble is white | Second marble is white) = ((1/2) * (3/7))/P(Second marble is white).
Now, we can solve for P(Second marble is white):
P(Second marble is white) = (1/2) * (3/7) + (5/12) * (4/7) = (6/14) + (20/84) = (36/84) + (20/84) = (56/84) = 2/3
Finally, substituting this value back into the equation above:
P(First marble is white | Second marble is white) = ((1/2) * (3/7))/(2/3) = (3/14) / (2/3) = (3/14) * (3/2) = 9/28.
Therefore, the probability that the first marble was white given that the second marble was white is 9/28.
prob(W from boxA) = 3/7
now we transfer a ball
case1, ball transferred is W
now we have 7white in B
prob(W from boxB) = 7/12
prob as stated = (3/7)(7/12) = 1/4
case 2, ball transferred is B
now we still have 6 W in B
prob(W from B) = 3/7 x 6/12 = 3/14
prob(of our event) = 1/4 + 3/14 = 13/28