Problem Solving

Needgmat 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

OAE

Please explain.

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
= E

RELATED VIDEOS
- The Complement: https://www.gmatprepnow.com/module/gmat ... /video/744
- Probability of Event A AND Event B: https://www.gmatprepnow.com/module/gmat ... /video/750

Hi Needgmat,

Probability 'math' allows you to calculate two things: what you WANT to have happen and what you DON'T WANT to have happen. Those two results always add up to 1, so sometimes the easiest way to answer a given probability question is to figure out what you DON'T WANT then subtract that fraction from 1 to figure out what you do WANT.

Here, we're told that the probability of an individual bulb failing is 0.06. We're told that a string of 10 light bulbs will fail if ANY of them fail. We're asked for the probability that the string of bulb will fail.

Trying to calculate every possible way that the string would fail would take a really, really long time. Since the GMAT will never give you a question that will require lots and lots of calculations, THAT approach cannot be the only way to answer the question. Instead, we should calculate the probability of what we DON'T WANT (that the string of bulbs functions correctly) and then subtract that from 1.

Probability that a bulb functions properly = 1 - 0.06 = 0.94

The probability that all 10 bulbs will function properly = (0.94)^10

The probability that the string of bulbs will fail = 1 - (0.94)^10

Final Answer: E

GMAT assassins aren't born, they're made,
Rich

Needgmat 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

OAE

We are given that, in a string of light bulbs, if any individual light bulb fails, the entire string fails. We need to find the probability that the string of light bulbs will fail during time period T. In other words we need to determine the probability that at least one light bulb fails.

It is instrumental in this problem to find the probability of the complement of the event of interest. If we find the probability that no light bulbs fail (i.e. that the string works properly), and then subtract that probability from 1, then we can easily find the probability that at least one light bulb fails.

Since P(at least one light bulb failing) + P(none of the 10 lightbulbs failing) = 1

P(at least one light bulb failing) = 1 - P(none of the 10 lightbulbs failing)

Thus, it's easiest to determine 1 - P(none of the 10 lightbulbs failing).

Since the probability that a light bulb will fail is 0.06 is, the probability that a light bulb won't fail is 1 - 0.06 = 0.94.

Thus, the probability that none of the 10 light bulbs fail is:

0.94 x 0.94 x 0.94 x 0.94 x 0.94 x 0.94 x 0.94 x 0.94 x 0.94 x 0.94

(0.94)^10

P(at least one light bulb failing) = 1 - (0.94)^10

Answer: E

Hi hoppycat,

Unfortunately, working through every possible way in which the string of lightbulbs could fail would take a really long time (and a lot of work). You would have to consider all of the ways in which one bulb would fail, two bulbs would fail, three bulbs would fail, etc. NONE of the questions that you'll face on the Official GMAT will require that amount of work to solve, so trying to do that work now would not be a good use of your time. You'd be better served learning the strategic/Tactical ways to avoid that type of work (which is what all of the Experts in this thread focused on).

GMAT assassins aren't born, they're made,
Rich

hoppycat wrote:I didn't use the complement law and ended up lost.
Can someone answer the question without that rule?

Hi hoppycat,

Well, computing the probability of failing is way too time-consuming and thus, not advisable. That cannot be dealt in less than 2 minutes.

You will have to go through this rigmarole to compute the probability of the string to fail.

Calculate the probability of the following and then add them.

1. Any one of the 10 light bulbs fails
2. Any two of the 10 light bulbs fail
3. Any three of the 10 light bulbs fail
.
.
.
.
.
.
10. All 10 light bulbs fail
---------------------

So, it's better that we find the probability of SUCCESS and deduct it from '1' to get above.

Hope this makes sense.

-Jay

Relevant book: Manhattan Review GMAT Combinatorics and Probability Guide
_________________
Manhattan Review GMAT Prep

Locations: New York | Vienna | Kuala Lumpur | Sydney | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.