Respuesta :
Answer:
The probability that we must wait 5 more minutes is 0.0000454
Step-by-step explanation:
Let X be the amount of minutes that passed between the two customers. X has emponential distribution with parameter [tex] \lambda = 1/0.5 = 2 [/tex] and its density is
[tex]f_X(x) = 2 \, e^{-2 \, x}[/tex]
We want to compute is P(X> 65 | X > 60), which is equal to P(X>65)/P(X>60) (Because the conditional event is contained in the other event). Lets do some computations
[tex]P(X>60) = \int\limits_{60}^{+\,\infty} {2e^{-2 \, x} } \, dx = - e^{-2x} \, |^{+\infty}_{\, 60} =e^{-120}[/tex]
[tex]P(X>65) = \int\limits_{65}^{+\,\infty} {2e^{-2 \, x} } \, dx = - e^{-2x} \, |^{+\infty}_{\, 65} = e^{-130}[/tex]
As a consequence
[tex]P(X > 65 | X > 60) = \frac{P(X > 65)}{P(X>60)} = \frac{e^{-130}}{e^{-120}} = e^{-10} = 0.0000454[/tex]
Note that this is the same result than just calculate P(X > 5), and 5 is, by the way, the difference between 60 and 65. This is commonly referred as the memoryless property: The past doesnt have influence in the future.
We conclude that the probability that we must wait 5 more minutes is 0.0000454.
