Here's how to solve the problem step-by-step:
1. Identify the Scenario:
We have four books, each of a different subject: physics, chemistry, mathematics, and biology. These books can be arranged in any order on a shelf.
2. Understand the Requirement:
We need to determine the probability that the mathematics book is on the top when the books are stacked randomly.
3. Counting the Total Number of Outcomes:
When stacking the four books, they can be arranged in any possible order. The total number of ways to arrange 4 distinct books is given by the factorial of 4, which is [tex]\( 4! \)[/tex].
[tex]\[
4! = 4 \times 3 \times 2 \times 1 = 24
\][/tex]
Thus, there are 24 possible ways to arrange the four books.
4. Favorable Outcomes:
For the mathematics book to be on the top, we consider the first position fixed with the mathematics book. The remaining three books (physics, chemistry, and biology) can be arranged in any order in the remaining three positions.
The number of ways to arrange the remaining three books is [tex]\( 3! \)[/tex].
[tex]\[
3! = 3 \times 2 \times 1 = 6
\][/tex]
Therefore, there are 6 favorable outcomes where the mathematics book is on top.
5. Calculate the Probability:
The probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes. Here, the number of favorable outcomes is 6, and the total number of possible outcomes is 24.
[tex]\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{6}{24} = \frac{1}{4}
\][/tex]
6. Conclusion:
Therefore, the probability that the mathematics book is on top is [tex]\( \frac{1}{4} \)[/tex].
So, the correct answer is [tex]\( B. \frac{1}{4} \)[/tex].
