MathProblem 76 Two bags and marble problem

You choose one of two identical looking bags at random. One bag has three black marbles and one white marble. The other has three white marbles and one black marble. After choosing a bag you draw one marble out at random. You notice it is black. You then put it back and draw another marble out of the same bag at random. What is the probability that the second marble drawn is black?

Solution

设事件 \(A: \text{second marble is black}; B:\text{first marble is black}\)，注意到是可放回的。所以利用贝叶斯公式

\[P(A|B)=\frac{P(AB)}{P(B)}=\frac{1/2\cdot1/4\cdot 1/4+1/2\cdot3/4\cdot3/4}{1/2\cdot1/4+1/2\cdot1/4}=\frac{5}{8} \]