# just a little help with random variables

• Nov 27th 2006, 06:38 PM
SwedishMan
just a little help with random variables
I can't figure this out. Help appreciated.

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 could really use some help. Thank you.
• Dec 1st 2006, 11:50 AM
WeeG
what you have to do is to define the probabilities of all the events:

P(X1) = 1/2 , because X1 will occur ( be 1 ) when the ball will have numbers 1 or 3 out of 4 options, meaning 2/4.
in the same way:
P(X2)=1/2
P(X3)=1/2

now you have to look at the intersections:

P(X1 and X2 ) = 1/4 (why ? ), and also P(X1)*P(X2)=1/4, so X1 and X2 are independent.
do the same for X1 & X3, X2 & X3.

the probability of all Xi to occur is 1/4, while P(X1)*P(X2)*P(X3)=1/8, so all the events are not independent.
you say that the 3 events are pairwise independent.