Probability of lights burning out in Series Circuit

[lth2525]

Hi, i have this question wrong in my intro statistics class and I'm wonder if someone could help me understand why. Thanks in advance.

Heres the question:

"A string of christmas lights contains 4 bulbs. The lights are wired in a series, so that if one fails, the whole sting would go dark. Each light has a probability of 0.05 of failing in a 3-year period. The lights fail independently of each other. What is the probability that a sting of lights will fail in a 3 year period?"

Now, my answer was 0.20, because each lights fail independetnly of each other, so then, if one fails, the others do not get a chance to fail. Since logically, w/o any external influence, a light bulb should not fail if it is not lit, therefore, 4 x .05 = 0.20. But i got the asnwer wrong, can someone help me to understand if I'm missing something? Thanks

On another note, can more than one light in a series circuit burn out at the same time?

 

[pig]

You can't add probabilities like that. Think about it, if you had 4 bulbs and each had a 0.25 probability of failing, you would get a 1 (100%) chance of failing for an answer and it is clear that this is not correct because it can still happen that none of the four fail. If you throw two coins, the chance that each will turn heads is 50%, but it's not 100% that at least one will

Try to think about it in a different way. What is the chance that a single lightbulb won't fail? Then, what is the chance that NONE of the bulbs will fail?

 

[HallsofIvy]

The probability of anyone bulb failing is 0.05 so the probability that anyone bulb WON'T fail is 0.95. The probability that NONE of the four lights will fail is (0.95)⁴= 0.81450625. Since the light string will fail is any of them does, that is the probability that the string will NOT fail so the probability that it will is 1-0.81450625= 0.18549375.

 

[lth2525]

O, Ok, thanks for explaining that to me, i understand it now