Question

In a string of Christmas lights, if any bulb fails, the entire string goes dark. Suppose a bulb has a 2% chance to fail before the end of the season. Further, assume that one bulb’s failure is independent of the others. If the string contains 20 bulbs, what is the probability the string fails during the holiday season (i.e. at least one of the bulbs fails)?

Answer 1

33.24%

Explanation:

To determine the probability which has the string to fail during the holiday season we will compute the probability of the string not failing and then subtract this quantity from 1.

Observe that:

• The string will work correctly only if each bulb does not fail.

Then, as the failure of a bulb is independent of the others, we have that the probability that every bulb does not fail is:

`(1-0.02)^(20)=(0.98)^(20)=0.6676=66.76\%.`

Therefore, the probability the string fails during the holiday season is given by 1-0.6676=0.3324=33.24%.