Question

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:

1. At least 20,000 hours

2. No longer than 4000 hours

3. Between 4,000 and 24,000 hours

Please show all work

Answer 1

Mean expected life = 20,000 hours

= 1 / Mean

= 0.00005

Ans 1)

At least 20,000 hours

The probability that one of these light bulbs will last at least 20,000 hours = e^{- * t}

= e^{- (0.00005 * 20000)}

= e^-1

= 0.368

Ans 2)

No longer than 4000 hours

The probability that one of these light bulbs will last no longer then 4,000 hours = 1 - e^{- * t}

= 1 - e^{-(0.00005 * 4000)}

= 1 - 0.818

= 0.181

Ans 3)

Between 4,000 and 24,000 hours

The probability that one of these light bulbs will last between 4,000 and 24,000 hours = 1 - (The probability that one of these light bulbs will last no longer then 4,000 hours + The probability that one of these light bulbs will last at least 24,000 hours)

= 1 - (0.181 + e^{-(0.00005 * 24000)})

= 1 - (0.181 + 0.301)

= 1 - 0.482

= 0.518