Probability that smoke detectors work

In summary, the conversation is discussing the probability of at least one smoke detector going off during a fire, given a 92% chance of each detector functioning properly. The conversation also mentions using the binomial distribution to solve the problem and suggests a more efficient approach by calculating the probability of none of the detectors going off and then applying it in a certain way.
  • #1
TSN79
424
0

Homework Statement



A smoke detector has a 92% chance of functioning properly.
If there are 3 of these, what are the chances at least one of them will go off during a fire?

Homework Equations



Standard binomial distribution I believe. X~B(n, p)

The Attempt at a Solution



P(X>=1) = P(X=3)+P(X=2)+P(X=1)
When placing this into the equation I end up with ~0.996.
Can someone verify?
 
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  • #2
I'm getting a different answer, you probably made a calculation error in there.

Consider: if you want to calculate the probability of at least one detector going off, it's easier to work 'the other way round'. You can calculate the probability that none of the smoke detectors will go off, and then apply that figure in a certain way to solve the problem. How exactly to do that is something I'll leave up to you to figure out for the moment! It's less work than what you did, in any case.

I believe you should arrive at an exact answer of 0.999488. Of course, take care in rounding off that figure to the correct number of decimals!
 
  • #3
What is the probability that one of them will not go off? What is the probability that none of them will go off? What is the probability that that will not happen?
 
  • #4
Brinx said:
I'm getting a different answer, you probably made a calculation error in there.

Are you saying I used correct procedure but made some arithmatic error?
 
  • #5
TSN79 said:
Are you saying I used correct procedure but made some arithmatic error?

Yes, I think so. When I calculate those separate chances (P(X=1) + P(X=2) + P(X=3)) I still arrive at the answer I mentioned. Could you post your calculations here? We'll be able to point out where things go wrong.
 
  • #6
Thx, I got the answer right now. Don't really know what I did wrong the first time...
 
  • #7
TSN79 said:
Thx, I got the answer right now. Don't really know what I did wrong the first time...

You're doing it in an inefficient way. Fine for 3 trials, but what if you're given 100 trials (say)? Will you manually add 100 terms?

Consider HallsofIvy's comment. That's the standard way to approach this sort of problem, saves a lot of work.
 

FAQ: Probability that smoke detectors work

What is the probability that a smoke detector will work when needed?

The probability that a smoke detector will work when needed depends on a variety of factors, including the quality of the detector, its location in the building, and whether or not it has been properly maintained. Generally, smoke detectors have a high probability of working when needed if they are installed and maintained correctly.

How accurate are smoke detectors in detecting fires?

Smoke detectors are designed to detect the presence of smoke, not fires specifically. They are highly accurate in detecting smoke particles in the air, which can indicate a fire. However, other factors such as cooking smoke or steam from a shower can also trigger a smoke detector, resulting in false alarms.

Do smoke detectors have a guarantee that they will work?

Most smoke detectors come with a manufacturer's guarantee or warranty that they will work for a certain period of time. However, it is important to regularly test and maintain your smoke detectors to ensure they are functioning properly. Even with a guarantee, there is always a small chance that a smoke detector may fail when needed.

Is the probability of a smoke detector working affected by its age?

Like any electronic device, smoke detectors can become less reliable as they age. It is recommended to replace smoke detectors every 10 years to ensure they are functioning properly. However, regular testing and maintenance can help prolong the lifespan of a smoke detector.

Can the probability of a smoke detector working be improved?

Yes, there are steps that can be taken to improve the probability of a smoke detector working when needed. These include regularly testing the detector, replacing the batteries at least once a year, and keeping the detector clean and free of dust and debris. It is also important to have multiple smoke detectors installed throughout the building to increase the chances of early detection of a fire.

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