Respuesta :
Answer:
43.62% probability that it will take more than an additional 34 minutes
Step-by-step explanation:
To solve this question, we need to understand the exponential distribution and the conditional probability formula.
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]
The probability of finding a value higher than x is:
[tex]P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^{-\mu x}) = e^{-\mu x}[/tex]
Conditional probability formula:
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]
In which
P(B|A) is the probability of event B happening, given that A happened.
[tex]P(A \cap B)[/tex] is the probability of both A and B happening.
P(A) is the probability of A happening.
In this question:
Event A: Taking more than 24 minutes.
Event B: Taking ore than 24+34 = 58 minutes.
P(A)
More than 24, use the exponential distribution.
Mean of 41, so [tex]m = 41, \mu = \frac{1}{41} = 0.0244[/tex]
[tex]P(A) = P(X > 24) = e^{-0.0244*24} = 0.5568[/tex]
Intersection:
More than 24 and more than 58, the intersection is more than 58. So
[tex]P(A \cap B) = P(X > 58) = e^{-0.0244*58} = 0.2429[/tex]
Then:
[tex]P(B|A) = \frac{0.2429}{0.5568} = 0.4362[/tex]
43.62% probability that it will take more than an additional 34 minutes
