High School What is the sum of multiple probabilities

[Observeraren]

If I have an asset that has a 10% chance to fail and I have ten of these assets in a basket, then what is the chance that one will fail in this basket? 10%? What is the chance of 10 failing? 0,01%?
Please also explain in some laymans terms. I am a total noob when it comes to mathematics.
This forum has helped me alot, thanks guys!

 

[mathman]

It is easier to start with the question: what is the probability that none will fail? Assuming the failures of these assets are independent of each other, then the probability that none will fail is q=0.9^{10}. The probability that at least one will fail =p=1-q.
Using independence again, the probability that all will fail is 0.1^{10}. I'll leave the arithmetic for you.

 

[Observeraren]

It is easier to start with the question: what is the probability that none will fail? Assuming the failures of these assets are independent of each other, then the probability that none will fail is q=0.9^{10}. The probability that at least one will fail =p=1-q.
Using independence again, the probability that all will fail is 0.1^{10}. I'll leave the arithmetic for you.

Taking into consideration that all these assets are independent from each other, is the probability that two will fail (p=1-q)/2?

So the probability that none will fail is q=0.3486784401%? and the probability that at least one will fail is p=0,65132156%?
This is magic to me I am not able to comprehend how this is possible that 10 of these assets with a 10% risk when summed up have a smaller risk than when alone.

Is there something I am not getting now? The probability that one will fail is not px100?

Wow.

 

[EngWiPy]

Having 10 assets failing simultaneously has a probability that is lower than having 1 asset failing (which makes sense, since intuitively it is less likely to happen). The general formula for having $1\leq n \leq 10$ assets fail simultaneously is $p^n$ since they are independent, where $p$ is the probability of failing.

 

[Observeraren]

Having 10 assets failing simultaneously has a probability that is lower than having 1 asset failing (which makes sense, since intuitively it is less likely to happen). The general formula for having $1\leq n \leq 10$ assets fail simultaneously is $p^n$ since they are independent, where $p$ is the probability of failing.

How nice of you to help me. p" role="presentation">p being the probability of one failing in this group as mathman posted above; p=1-q?

This is such magic to me, how can the fail probability of 1/10 of these assets be less than 10%? p=1-q. Or does p=.65... mean that there is a 65% probability that one will fail? that seems logical.

I have come to the conclusion that my latter statement is the case. 1.00 being 100%.
probabiliies are still magic tho.
wow.

 

[FactChecker]

If I have an asset that has a 10% chance to fail and I have ten of these assets in a basket, then what is the chance that one will fail in this basket? 10%?

Maybe. Do you mean given a specific one, that it will fail? Or do you mean that out of all ten exactly one will fail? If you mean the latter, then 10% is wrong.

For exactly one to fail out of all ten, calculate the sum of #1 fail and #2-#10 pass + #2 fail and #1,#3-#10 pass + ... + #10 fail and #1-#9 pass. Since all these cases are disjoint you can simply add the probabilities.

 

[WWGD]

How nice of you to help me. p" role="presentation">p being the probability of one failing in this group as mathman posted above; p=1-q?

This is such magic to me, how can the fail probability of 1/10 of these assets be less than 10%? p=1-q. Or does p=.65... mean that there is a 65% probability that one will fail? that seems logical.

.

Note that in one case you compute the probability of _either_ asset1, asset2,..., asset10 failing and not just asseti failing ; I=1,2,..,10.

 

[benorin]

The And Rule: If events A and B are independent, then P(A and B) = P(A)⋅P(B).

Since P(A) and P(B) are probabilities we have 0 ≤ P(A), P(B) ≤ 1 ⇒ P(A and B) = P(A)⋅P(B) ≤ P(A), P(B) because when u multiply two numbers between 0 and 1 the result is always smaller than either number you started with. So the probability of having two independent events happen together is always less than or equal to the probability of either event on it's own.

Hope that helps.

 

[sysprog]

How nice of you to help me. p" role="presentation">p being the probability of one failing in this group as mathman posted above; p=1-q?

This is such magic to me, how can the fail probability of 1/10 of these assets be less than 10%? p=1-q. Or does p=.65... mean that there is a 65% probability that one will fail? that seems logical.

I have come to the conclusion that my latter statement is the case. 1.00 being 100%.
probabiliies are still magic tho.
wow.

That is correct --
others have already pointed out some fundamentals --
as you have concluded for this example:
0.9^10 is 0.348678, and 100-34.8678 puts the chance of at least one asset failing at 65.1322%
The chance that all 10 will fail is 0.1^10, or 1E-10,
so the chance that at least one will not fail is 1 - 1E-10, which is 99.99999999%