Bag A has $3$ white and $2$ black marbles. Bag B has $4$ white and $3$ black marbles.
Suppose we draw a marble at random from Bag A and put it in Bag B. After doing this, we draw a marble at random from Bag B, which turns out to be white. Given this information, what is the probability that the marble we moved from Bag A to Bag B is white?
This problem is different from other conditional probability problems. It has a changing variable. I'm stuck on how to approach this problem. Could someone pelase walk me step by step through this problem? Thanks!
Answer
The law of total probability helps here. If the transferred marble was white, which happens $\frac35$ of the time, the probability that the marble drawn from bag B is white is $\frac58$. If the transferred marble was black ($\frac25$ chance), that probability is $\frac12$.
The probability that the drawn marble is white and the transferred marble is white is $\frac58×\frac35=\frac38$. The probability that the drawn marble is white but the transferred marble is black is $\frac12×\frac25=\frac15$. Therefore the probability the transferred marble was white given that the drawn marble is white is
$$\frac{\dfrac38}{\dfrac38+\dfrac15}=\frac{15}{23}$$
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