Answer to Question #211331 in Statistics and Probability for Naresh

Question

A fair six-sided dice is thrown and the scores are noted.

Event X: The total of the two scores is 4.

Even Y: The first score is 2 or 5.

a) Are events X and Y independent

Answer 1

Solution:

We assume the die is thrown twice.

X = {(1,3),(3,1),(2,2)

Y = {(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)}

Now, X $\cap$ Y = {(2,2)} $\ne \phi$

And we know that if A and B are two independent events, then $P(A).P(B)=P(A\cap B)$

Here, $P(X)=\dfrac{3}{36}=\dfrac{1}{12},P(Y)=\dfrac{12}{36}=\dfrac{1}{3}$

And $P(X\cap Y)=\dfrac{1}{36}$

Clearly, $P(X).P(Y)=P(X\cap Y)=\dfrac{1}{36}$

So, events $X, \,Y$ are independent.