# Probability

#### quaidy4

##### New member
A string of Christmas lights has 20 bulbs and if any bulb fails then the whole string goes out.
Suppose that each bulb has a 5% chance of failure during the Christmas holidays and that
the 20 bulbs are independent of each other.
What is the probability that the string of lights will go out during the holidays?

Would you do a combination formula first, and then do the complement formula? or how would you approach this.

#### soroban

##### Elite Member
Hello, quaidy4!

A string of Christmas lights has 20 bulbs and if any bulb fails then the whole string goes out.
Suppose that each bulb has a 5% chance of failure during the Christmas holidays
and that the 20 bulbs are independent of each other.
What is the probability that the string of lights will go out during the holidays?

Would you do a combination formula first, and then do the complement formula? . Yes!

$$\displaystyle \text{For each bulb, we have: }\;\begin{Bmatrix} P(\text{fail}) &=& 0.05\\ \\[-4mm] P(\text{go\,\!o\,\!d}) &=& 0.95 \end{Bmatrix}$$

The opposite of "at least one bulb fails" is "all 20 bulbs are good."

. . $$\displaystyle P(\text{all 20 good}) \;=\;\left(0.95\right)^{20}\;=\;0.358485922$$

$$\displaystyle \text{Therefore: }\(\text{at least one fails}) \;\;=\;\;1 \,-\, 0.358485922 \;\;=\;\;0.641514078 \;\;\approx\;\;64\%$$

#### quaidy4

##### New member
okay! thanks so much!