Thread: Probability That None of Independent Events Occurs

The problem says:

Let be independent events. Show that the probability that none of the occur is less than or equal to exp(- .

I'm confused at where the inequality and the exponent are coming from.

Here is what i've done.

I figured that the P(none happening) = 1 - P(one happening)

P(one happening) = .

Then, P(none) = 1 - .

I think this is right, but where to go from here, I'm not sure.

Thank you!

First of all
P(none happening) = 1 - P(at least one happening)

Then P(at least one happening)



BUT by independence this is an equality (which is why it works).

What you seem to need is that

Use calculus to prove that.

Let

then obtain the first two derivatives and show that the min is 0 at 0, so .

So, the only thing I had wrong was that it should be P(at least one happening) rather than just P(one happening), but that should still be equal to , correct?

I will try to work out the other stuff today. Thank you!

That is correct.

Now all you need to show is that



which is equivalent to proving, as matheagle explained:



EDIT: Thanks matheagle, I'm always messing up =X

you're missing the minus in the exponent
but that's true too



we want

Originally Posted by pedrosorio

That is correct.

Now all you need to show is that



which is equivalent to proving, as matheagle explained:

Thank you both very much! I got it now