On 2 shelves there are biology and geography books. On the first
shelf, there are 6 biology and 4 geography books, and on the second
one, there are 5 biology and 7 geography books. The student from the
first shelf takes 2 books and puts them on the desk and from the
second one takes 1 book and puts it on the desk. In the end, the
student takes 1 book from the desk. What is the probability that it
is a biology book?
I have several questions regarding this problem: The probability of getting a biology book from the first shelf is 60%. But what happens when we take two books? It looks to me that the probability should go up, but how do we calculate it?
And if we calculate the probability for each shelf how do we "combine" the probabilities to get the final result?
Best Answer
Every biology book on shelf 1 has probability $\frac{2}{10}\frac{1}{3}=\frac{1}{15}$ to end up as the book that is finally taken by the student from the desk.
Every biology book on shelf 2 has probability $\frac{1}{12}\frac{1}{3}=\frac{1}{36}$ to end up as the book that is finally taken by the student from the desk.
So the probability that a biology book is finally taken by the student from the desk is: $$\frac{6}{15}+\frac{5}{36}$$
If $2$ books are taken from shelf $1$ then the probability that no biology books are taken equals $\frac4{10}\frac39=\frac{12}{90}$ so that the probability at least one biology book is taken equals $1-\frac4{10}\frac39=\frac{78}{90}$.
The probability that exactly $1$ biology book is taken from shelf 1 equals $2\times\frac6{10}\frac49=\frac{48}{90}$.
The probability that exactly $2$ biology books are taken from shelf 1 equals $\frac6{10}\frac5{9}=\frac{30}{90}$.