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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?

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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 $Y=0$.)

b) Do the same analysis for your other cases.

Yes... you can also check this through:

Is $P(X|Y) = P(X)$.

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