Understanding Independence in Probability: Solving a Class Problem

[JaysFan31]

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?

 

[mighty2000]

it is important to understand the concepts of independence and how it applies to random variables. In this case, the mistake lies in the assumption that all combinations of the three random variables have equal probabilities.

To show that the three random variables are not independent, we need to find at least one combination where the joint probability does not equal the product of individual probabilities. Let's look at the combination where X1=1, X2=1, and X3=1. This means that ball 1 is drawn. The joint probability of this combination is 1/4, since there is only one ball numbered 1 in the box. However, the product of individual probabilities is (1/2)*(1/2)*(1/2)=1/8, which is not equal to 1/4.

This shows that the three random variables are not independent, as the joint probability does not equal the product of individual probabilities for at least one combination. Therefore, it is incorrect to conclude that all three random variables are independent based on the assumption of equal probabilities for all combinations.

In order to properly show independence between random variables, it is important to consider all possible combinations and calculate the joint probability for each one. Only when the joint probability equals the product of individual probabilities for all combinations, can we conclude that the random variables are independent.