Question

A fair sided 6-sided die is tossed twice- the number on the upface is recorded after each toss. Let y1=the number of appearances of a six in the two tosses and let y2= the number of appearances of an odd number in two tosses. Find the joint probability for y1 and y2. Give it as a table

Answer 1

The joint probability distribution of Y₁ and Y₂ here is given as:

	Y1 = 0	Y1 = 1	Y1 = 2
Y2 = 0	(2/6)2 = 4/36	2*(1/6)*(2/6) = 4/36	(1/6)2 = 1/36
Y2 = 1	2*(2/6)*(3/6) = 12/36	2*(3/6)*(1/6) = 6/36	0
Y2 = 2	(3/6)2 = 9/36	0	0

This is the required joint PDF for Y₁ and Y₂.

Note that the above probabilities are obtained as follows:

P(Y₁ = Y₂ = 0) = Probability that both tosses have got a number from {2, 4}
P(Y₁ = 0, Y₂ = 1) = Probability that there is an odd number in one of the 2 tosses and a number from {2, 4} on the other toss
P(Y₁ = 0, Y₂ = 2) = Probability that there is an odd number on both tosses
P(Y₁ = 1, Y₂ = 0) = Probability of having a six on one of the dice and one of {2, 4} on the other
and so on....