To solve this problem, we need to determine the probability that a randomly thrown rock hits one of the cracked pieces of glass in the window. The steps to find this probability are as follows:
1. Identify the Total Number of Pieces: The window is made up of 15 equal-sized pieces of glass. Thus, the total number of pieces in the window is 15.
2. Identify the Number of Cracked Pieces: Among these 15 pieces, 3 are cracked.
3. Define the Probability Formula: The probability of an event occurring is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. In this case, the favorable outcomes are hitting a cracked piece, and the possible outcomes are hitting any piece of the window. Mathematically, it is expressed as:
[tex]\[
P(\text{cracked}) = \frac{\text{number of cracked pieces}}{\text{total number of pieces}}
\][/tex]
4. Plug in the Values: Substitute the number of cracked pieces (3) and the total number of pieces (15) into the formula:
[tex]\[
P(\text{cracked}) = \frac{3}{15}
\][/tex]
5. Simplify the Fraction: Simplify the fraction [tex]\(\frac{3}{15}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
[tex]\[
\frac{3}{15} = \frac{3 \div 3}{15 \div 3} = \frac{1}{5}
\][/tex]
Thus, the probability that the rock hits a cracked piece of glass is [tex]\(\frac{1}{5}\)[/tex].
Therefore, the correct answer is:
B. [tex]\(\frac{1}{5}\)[/tex]
