Question

question 1 of 5
a string of christmas lights contains 20 lights. the lights are wired in series, so that if any light fails, the whole string will go
dark. each light has probability 0.98 of working for a 3 - year period. the lights fail independently of each other.
find the probability that the string of lights will remain bright for 3 years.
(round to 3 decimal places. leave your answer in decimal form.)

Explanation:

Step1: Determine the probability formula

Since the lights are independent, the probability that all \(n = 20\) lights work is \(P=(p)^{n}\), where \(p = 0.98\) is the probability that a single light works.

Step2: Calculate the probability

Substitute \(n = 20\) and \(p=0.98\) into the formula: \(P=(0.98)^{20}\).
Using a calculator, \((0.98)^{20}=e^{20\ln(0.98)}\) (by the property \(a^{b}=e^{b\ln(a)}\)).
\(\ln(0.98)\approx - 0.020202706\), then \(20\ln(0.98)\approx20\times(- 0.020202706)=-0.40405412\).
\(e^{-0.40405412}\approx0.667607\)

Answer:

\(0.668\)