Question

Find the probability that urn j and urn k are empty

Answer 1

I already figured out the first and the last part but not sure about the second one. Hope someone would help me clarify it, thanks!

Answer 2

@oldrin.bataku You have any idea?

Answer 3

Okay, so the 3 possible tuples are (j,k)=(2,3) or (2,4) or (3,4). I've computed the probability corresponding to each of them but does the probability I was asked for (P{I_j=1,I_k=1}) equal to the summation of them or I must condition on other variables?

Answer 4

They are mutually exclusive, and the events OR'd together get the target event, no?

Answer 5

Wait, they aren't mutually exclusive...

Answer 6

You also need to calculate the probability of 2, 3, and 4 begin empty to eliminate double counting.

Answer 7

P{I_3=1, I_4=1}=1/3, P{I_2=1, I_4=1}=1/6, P{I_2=1, I_4=1}=1/12 right? What's to do next?

Answer 8

I don't actually know if that is right. I'm haven't done the problem and don't know your reasoning.

Answer 9

These are the events for empty urns: \[
A=(2,3), B=(2,4), C=(3,4), D=(2,3,4)
\]This is the probability for which we are looking:\[
\Pr(A\cup B\cup C)
\]So we start with \[
\Pr(A\cup B\cup C) = \Pr(A\cup B)+\Pr(C)-\Pr((A\cup B)\cap C)
\]Then remember that \[
\Pr(A\cup B) = \Pr(A)+\Pr(B)-\Pr(A\cap B)
\]Lastly is to find \[
\Pr((A\cup B)\cap C)
\]

Answer 10

Remember that \(((A\cup B)\cap C)=D\).
\((A\cup B)\) ensure that \(2\) is empty. \(C\) ensure that \(3,4\) are empty.

Answer 11

Another note is that \(A\cap B=D\) as well, for similar reasoning.

Answer 12

Well, thanks for it. So I should interpret the problem as taking the union, not the summation of the events. :)

Answer 13

Union represent OR.
Summation only applies for mutually exclusive events... these ones aren't.