Probability Question

J

JaysFan31

Guest
I have the following problem for one of my classes:

A box contains four balls, numbered 1 through 4. One ball is selected at random from this box.
Let X1=1 if ball number 1 or ball number 2 is drawn.
Let X2=1 if ball number 1 or ball number 3 is drawn.
Let X3=1 if ball number 1 or ball number 4 is drawn.
The Xi values are zero otherwise. Show that any two of the random variables X1, X2, X3 are independent but that the three together are not.

I was able to show that any two of X1, X2, X3 are independent according to the following definition:
p(Y1, Y2)=p(Y1)p(Y2).
I did the following work:
I said that for just two random variables there are sixteen combinations. Also the p(X1)=p(X2)=p(x3)=0.50.
Thus, for (X1, X2)
p(X1, X2)=p(X1)p(X2)
p(X1, X2)=p(1,1)+p(1,3)+p(2,1)+p(2,3)
Since there are sixteen combinations, each individual one has probability of 1/16. Therefore, p(X1, X2)=0.25, which is equal to p(X1)*p(X2) since each of these are 0.5. I followed this same pattern for all combinations of random variables and they all worked. Thus, I concluded that any two of the random variables are independent.

However, I did the same thing for all three and got that they should be independent:
There are 64 combinations with 3 random variables.
P(X1, X2, X3) just quick appears to equal 8/64=1/8.
But so does p(X1)*p(X2)*p(X3) since (0.5)^3=1/8.
I should be able to conclude that all three of them are NOT independent, but I can't using my method.

Where is my mistake?
 

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