# Discrete Probability: Random Variable Independent or Dependent?

Question a):

You flip a fair coin $$7$$ times; these coin flips are independent of each other.

Define the random variables:

$$X$$ = the number of heads in these $$7$$ coin flips, and

$$Y$$ = the number of tails in these $$7$$ coin flips.

Are the random variables independent or not? Show why or why not?

Attempt:

I know there are $$2^{7}$$ = $$128$$ possible sequences for the con toss $$7$$ times.

{HHHHHHT, HHHHHTT, HHHHTTT, HHHTTTT, HHTTTTT, HTTTTTT, HHHHHHH}

If I take $$P(X=1$$ and $$Y=1) = P(X=1)P(Y=1)$$

The Probability of just 1 head and 1 tail would be $$\frac{7}{128}$$ for both.

The Probability of just 1 head and 1 tail occurring together is $$0$$

So, they are not equal and hence dependent. Is this correct way?

Question b):

Consider the set $$(1,2,3,…10)$$. You choose a uniformly random element z in S.

Define the random variables:

$$X$$ = $$0$$ if z is even and $$1$$ is z odd

$$Y$$ = $$0$$ if z {1,2}, $$1$$ if z {3,4,5,6}, $$2$$ if z {7,8,9,20}

Are the random variables independent or not? Show why or why not?

Attempt:

$$Pr(X = 0$$ or $$1)$$ for both odd and even is $$2^{5} / 2^{10}$$ = $$\frac{1}{32}$$.

$$Pr(Y = 0) =$$ $$1$$ subset {1,2} / $$2^{10}$$ possible subsets

$$Pr$$($$X=0$$ and $$Y=0$$) = $$P$$($$X=0)P(Y=0)$$

$$\frac{1}{2^{10}}$$ $$.$$ $$\frac{1}{32}$$. = $$\frac{1}{32}$$ $$.$$ $$\frac{1}{2^{10}}$$

They are equal and hence independent. Is my approach to finding the probabilities correct to come to this conclusion of independence?

The way to understand independence is to ask yourself: "If I know that variable $$X$$ has a particular value, does that provide any information whatsoever about the value of variable $$Y$$?"
a) So imagine $$X = 7$$. Does that provide any information whatsoever about the value of variable $$Y$$? Of course it does! You can be certain that $$Y \neq 7$$, for instance. (We can only have $$Y=0$$.)
Is $$P(X|Y) = P(X)$$.