Answer:
0.069 = 6.9% probability that a customer has to wait more than 4 minutes.
Step-by-step explanation:
Exponential distribution:
The exponential probability distribution, with mean m, is described by the following equation:
[tex]f(x) = \mu e^{-\mu x}[/tex]
In which [tex]\mu = \frac{1}{m}[/tex] is the decay parameter.
The probability that x is lower or equal to a is given by:
[tex]P(X \leq x) = \int\limits^a_0 {f(x)} \, dx[/tex]
Which has the following solution:
[tex]P(X \leq x) = 1 - e^{-\mu x}[/tex]
In this problem, we have that:
[tex]m = 1.5[/tex]
So
[tex]\mu = \frac{1}{1.5} = 0.6667[/tex]
[tex]P(X \leq x) = 1 - e^{-0.667x}[/tex]
Find the probability that a customer has to wait more than 4 minutes.
Either the customer has to wait 4 minutes or less, or he has to wait more than 4 minutes. The sum of the probabilities of these events is decimal 1. So
[tex]P(X \leq 4) + P(X > 4) = 1[/tex]
We want P(X > 4). So
[tex]P(X > 4) = 1 - P(X \leq 4) = 1 - (1 - e^{-0.667*4}) = 0.069[/tex]
0.069 = 6.9% probability that a customer has to wait more than 4 minutes.
