Mathematics: Lucky Lumen light bulbs have an expected life that is exponentially distributed with a mean of 20,000 hours. Determine the probability that one of these light bulbs will last a. At least 24,000 hours. b. No longer than 4,000 hours. c. Between 4,000 hours and 24,000 hours.

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Lucky Lumen light bulbs have an expected life that

bobgardner765

17.03.2021

a) 0.3012 = 30.12% probability that one of these light bulbs will last at least 24,000 hours.

b) 0.1813 = 18.13% probability that one of these light bulbs will last no longer than 4,000 hours.

c) 0.5175 = 51.75% probability that one of these light bulbs will last between 4,000 hours and 24,000 hours.

Step-by-step explanation:

Exponential distribution:

The exponential probability distribution, with mean m, is described by the following equation:

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

In which $\mu = \frac{1}{m}$ 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:

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

Mean of 20,000 hours.

This means that $m = 20000, \mu = \frac{1}{20000} = 0.00005$

a. At least 24,000 hours.

$P(X 24000) = e^{-0.00005*24000} = 0.3012$

0.3012 = 30.12% probability that one of these light bulbs will last at least 24,000 hours.

b. No longer than 4,000 hours.

$P(X \leq 4000) = 1 - e^{-0.00005*4000} = 0.1813$

0.1813 = 18.13% probability that one of these light bulbs will last no longer than 4,000 hours.

c. Between 4,000 hours and 24,000 hours.

Less than 4000 or more than 24000:

We found in a and b, so

0.3012 + 0.1813 = 0.4825

Between these bounds:

1 - 0.4825 = 0.5175

0.5175 = 51.75% probability that one of these light bulbs will last between 4,000 hours and 24,000 hours.

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