A string of 10 light bulbs is wired in such a way that if an

[ghong14]

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A string of 10 light bulbs is wired in such a way that if any individual light bulb fails, the entire string fails. If for each individual light bulb the probability of failing during time period T is 0.06, what is the probability that the string of light bulbs will fail during the time period T?

A. 0.06
B. (0.06)^10
C. 1 - (0.06)^10
D. (0.94)^10
E. 1 - (0.94)^10

If the probability of any of the light bulb failing is .06. Then why do we need to calculate the probability of each of the light bulbs not failing and subtract that from 1. Didn't really see the logic in this.

[jlucero]

Because there's a lot of ways that the bulbs could fail, but only one way that the bulbs don't. In order to solve this traditionally, we'd have to find the probabilities that 1, 2, 3.... or all 10 bulbs fail and add those separate probabilities together. That's a lot of work. Or since there's a 100% likelihood that somewhere between 0-10 bulbs will fail, if we find the probability that none of the bulbs fail, we can subtract that from 1 to find the probability that one of the bulbs does fail. We call this the 1-x trick and is similar to finding the odds of losing the lottery. If you have a 1 in a million chance of winning the lottery, you subtract that from 1 (100%) to find the probability you don't win: 999,999 out of a million.

[ghong14]

I understand the 1-x trick However if we know that the probability that any one of them may fail is .06 then isn't that the probability since it is only going to take 1 failure for the lights to go off? Are u saying that this does not include if 2 or 3 of them fail at the same time?

If that is the case then we would need 1-probability that one will not fail which is .94. However, we need to make sure that none of the 10 bulbs fail so it would be .94x.94x.94....=.94^10. Now if we subtract 1-.94^10 then we should get the probability that if any of the bulbs fail including 2 or three failures at a time.

Is that the right concept?

[RonPurewal]

I understand the 1-x trick However if we know that the probability that any one of them may fail is .06 then isn't that the probability since it is only going to take 1 failure for the lights to go off?

when you are puzzling through questions like this, you should see whether your question still makes sense in another case. say, with much bigger numbers.
what if there were a million bulbs on the string? in this case, common sense dictates that the probability of failure is going to be very, very close to 100%, because ... well, because, let's be serious here, all one million of your bulbs are not going to be perfect.

with this logic, though, you'd still think the probability is only 0.06 even with a million (or a billion or a quadrillion) bulbs. so, time to toss that reasoning -- it doesn't pass the smell test.

Are u saying that this does not include if 2 or 3 of them fail at the same time?

* right, it doesn't include those cases.
* it also doesn't include any of the other individual bulbs.
see, your 0.06 is just the probability that the first bulb fails... but that's not even taking into account the possibility that any of the other nine bulbs fails (and then, on top of that, the possibility of multiple failures).

If that is the case then we would need 1-probability that one will not fail which is .94. However, we need to make sure that none of the 10 bulbs fail so it would be .94x.94x.94....=.94^10. Now if we subtract 1-.94^10 then we should get the probability that if any of the bulbs fail including 2 or three failures at a time.

Is that the right concept?

that's the right concept. (and it includes not only 1, 2, or 3 failures at the same time, but also 4 or 5 or 6 or 7 or 8 or 9 or 10 failures at the same time.)