To determine the probability that the kayak rental company has enough kayaks on any given day, we can model the number of kayaks rented out each day as a normal distribution.
Given:
- Mean (μ) = 42 kayaks per day
- Standard deviation (σ) = 4 kayaks
- Number of kayaks available = 45
We want to find the probability that the number of kayaks rented out (X) is less than or equal to 45:
\[ P(X \leq 45) \]
To find this probability, we first need to standardize the value using the z-score formula:
\[ z = \frac{X - \mu}{\sigma} \]
Substituting the values we have:
\[ z = \frac{45 - 42}{4} = \frac{3}{4} = 0.75 \]
Next, we look up the z-score of 0.75 in the standard normal distribution table or use a calculator or spreadsheet to find the corresponding probability.
Using a standard normal distribution calculator, we find:
\[ P(Z \leq 0.75) \approx 0.7734 \]
This means that the probability of renting out 45 or fewer kayaks on any given day is approximately 0.7734.
Rounding this to the nearest tenth, we have:
\[ \text{Probability that there are enough kayaks available} \approx 0.8 \]
Thus, the probability that the company will have enough kayaks on any given day is 0.8 (or 80%).