Question

A Christmas tree light has an expected life of 200 hr and a standard deviation of 2 hr.

(a) Find a bound on the probability that one of these Christmas tree lights will require replacement between 190 hr and 210 hr.
at least ? %

(b) Suppose a large city uses 160,000 of these Christmas tree lights as part of its Christmas decorations. Estimate the number of lights that are likely to require replacement between 180 hr and 220 hr of use.
? lights

Answer 1

(a)

Probability that one of these Christmas tree lights will require replacement between 190 hr and 210 hr = P(|X - E[X]| 10)

By Chebyshev's inequality,

P(|X - E[X]| 10) Var(X) / 10²

=> P(|X - E[X]| 10) 2² / 10²

=> P(|X - E[X]| 10) 0.04

=> 1 - P(|X - E[X]| 10) 0.04

=> P(|X - E[X]| 10) 0.96

Thus, probability that one of these Christmas tree lights will require replacement between 190 hr and 210 hr is at least 0.96.

(b)

Probability that one of these Christmas tree lights will require replacement between 180 hr and 220 hr = P(|X - E[X]| 20)

By Chebyshev's inequality,

P(|X - E[X]| 20) Var(X) / 20²

=> P(|X - E[X]| 20) 2² / 20²

=> P(|X - E[X]| 20) 0.01

=> 1 - P(|X - E[X]| 20) 0.01

=> P(|X - E[X]| 20) 0.99

Thus, probability that one of these Christmas tree lights will require replacement between 180 hr and 220 hr is at least 0.99.

Number of lights that are likely to require replacement between 180 hr and 220 hr of use = 160,000 * 0.99

= 158400