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A string of 10 lightbulbs is wired in such a way that if any

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Carcass
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A string of 10 lightbulbs is wired in such a way that if any [#permalink] New post   07 Sep 2020, 04:45
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A string of 10 lightbulbs is wired in such a way that if any individual lightbulb fails, the entire string fails. If for each individual lightbulb the probability of failing during time period T is 0.06, what is the probability that the string of lightbulbs will fail during time period T ?


A. 0.06

B. (0.06)10

C. 1−(0.06)10

D. (0.94)10

E. 1−(0.94)10
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E
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Re: A string of 10 lightbulbs is wired in such a way that if any [#permalink] New post   07 Sep 2020, 06:00
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Carcass wrote:
A string of 10 lightbulbs is wired in such a way that if any individual lightbulb fails, the entire string fails. If for each individual lightbulb the probability of failing during time period T is 0.06, what is the probability that the string of lightbulbs will fail during time period T ?


A. 0.06

B. (0.06)10

C. 1−(0.06)10

D. (0.94)10

E. 1−(0.94)10


Aside: If P(bulb fails) = 0.06, then P(bulb doesn't fail) = 0.94

Okay, the entire string of lightbulbs will fail if 1 or more lightbulbs fail.
So, we want P(at least 1 lightbulb fails)

When it comes to probability questions involving "at least," it's best to try using the complement.
That is, P(Event A happening) = 1 - P(Event A not happening)
P(at least 1 lightbulb fails) = 1 - P(zero lightbulbs fail)


P(zero lightbulbs fail)
P(zero lightbulbs fail) = P(1st bulb doesn't fail AND 2nd bulb doesn't fail AND 3rd bulb doesn't fail AND . . . AND 9th bulb doesn't fail AND 10th bulb doesn't fail)
= P(1st bulb doesn't fail) x P(2nd bulb doesn't fail) x P(3rd bulb doesn't fail) x . . . x P(9th bulb doesn't fail) x P(10th bulb doesn't fail)
= (0.94) x (0.94) x (0.94) x . . . x(0.94) x (0.94)
= (0.96)^10

So, P(at least 1 lightbulb fails) = 1 - P(zero lightbulbs fail)
= 1 - (0.94)^10

Answer: E

Cheers,
Brent
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Re: A string of 10 lightbulbs is wired in such a way that if any [#permalink] New post   07 Sep 2020, 04:46
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Re: A string of 10 lightbulbs is wired in such a way that if any [#permalink] New post   20 Jun 2023, 13:52
Exhausting all the options and combinations (from 1 bulb failing to all bulb failing) would likely take you 10+ minutes - not a wise idea when you have 35 minutes for 20 questions.
We can decompose the probabilities like this:
We are told that if even just one bulb fails, the entire string fails.
Therefore:
P(at least one fails)+P(no failures)=1
P(at least one fails)=1-P(no failures)
P(no failure) is just the probability that all the bulbs dont fail which is given to us in the stem:
P(fail)=0.06, implying that P(no fail)=0.94
So p(no failure)=0.94^10,
and P(at least one fails)=1-(0.94^10)
which is exactly option E
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Re: A string of 10 lightbulbs is wired in such a way that if any [#permalink] New post   16 Jul 2024, 11:47
Hello from the GRE Prep Club BumpBot!

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Re: A string of 10 lightbulbs is wired in such a way that if any [#permalink]
16 Jul 2024, 11:47
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