An exponential distribution is formed by the time it takes for a person to choose a birthday gift. The average time it takes for a person to choose a birthday gift is 41 minutes…

Question

asked Aug 21, 2021 93.4k views

1 Answer

Jason Shultz · Answer 1 · 2021-08-27T18:36:24+0000

Answer:

43.62% probability that it will take more than an additional 34 minutes

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:

$f(x) = \mu e^(-\mu x)$

In which
$\mu = (1)/(m)$ is the decay parameter.

The probability that x is lower or equal to a is given by:

$P(X \leq x) = \int\limits^a_0 {f(x)} \, dx$

Which has the following solution:

$P(X \leq x) = 1 - e^(-\mu x)$

The probability of finding a value higher than x is:

$P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^(-\mu x)) = e^(-\mu x)$

Conditional probability formula:

$P(B|A) = (P(A \cap B))/(P(A))$

In which

P(B|A) is the probability of event B happening, given that A happened.

$P(A \cap B)$ 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
$m = 41, \mu = (1)/(41) = 0.0244$

P(A) = P(X > 24) = e^(-0.0244*24) = 0.5568

Intersection:

More than 24 and more than 58, the intersection is more than 58. So

$P(A \cap B) = P(X > 58) = e^(-0.0244*58) = 0.2429$

Then:

P(B|A) = (0.2429)/(0.5568) = 0.4362

43.62% probability that it will take more than an additional 34 minutes