To determine the expected number of motor scooters rented out in a month, we need to calculate the expected value (mean) using the given probabilities for each possible number of rented scooters.
The formula for the expected value [tex]\( E(X) \)[/tex] where [tex]\( X \)[/tex] is a discrete random variable is:
[tex]\[ E(X) = \sum_{i=1}^{n} x_i \cdot P(x_i) \][/tex]
Here, [tex]\( x_i \)[/tex] is each possible value of [tex]\( X \)[/tex] (in our case, the number of scooters), and [tex]\( P(x_i) \)[/tex] is the corresponding probability.
Given the data:
- Number of scooters ([tex]\( x_i \)[/tex]): 0, 1, 2, 3, 4, 5
- Probabilities ([tex]\( P(x_i) \)[/tex]): [tex]\( \frac{1}{32} \)[/tex], [tex]\( \frac{5}{32} \)[/tex], [tex]\( \frac{10}{32} \)[/tex], [tex]\( \frac{10}{32} \)[/tex], [tex]\( \frac{5}{32} \)[/tex], [tex]\( \frac{1}{32} \)[/tex]
Now, let's calculate the expected value step by step:
1. Multiply each number of scooters by its corresponding probability:
- [tex]\( 0 \cdot \frac{1}{32} = 0 \)[/tex]
- [tex]\( 1 \cdot \frac{5}{32} = \frac{5}{32} \)[/tex]
- [tex]\( 2 \cdot \frac{10}{32} = \frac{20}{32} \)[/tex]
- [tex]\( 3 \cdot \frac{10}{32} = \frac{30}{32} \)[/tex]
- [tex]\( 4 \cdot \frac{5}{32} = \frac{20}{32} \)[/tex]
- [tex]\( 5 \cdot \frac{1}{32} = \frac{5}{32} \)[/tex]
2. Sum these products:
[tex]\[
0 + \frac{5}{32} + \frac{20}{32} + \frac{30}{32} + \frac{20}{32} + \frac{5}{32}
= \frac{5 + 20 + 30 + 20 + 5}{32}
= \frac{80}{32}
= 2.5
\][/tex]
Thus, the expected number of motor scooters to be rented out in a month is [tex]\( 2.5 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{2.5} \][/tex]
