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Question:

a) Let $n$ $ge$ $3$ be an integer. Consider a uniformly random permutation $a_1$,$a_2$...$a_n$ of the set ${(1,2,...,n})$. Define the events:

A = "$a_n$ = n"

B = "$a_2$ $gt$ $a_1$"

Are these independent or dependent? Show why?

Attempt:

I have to show $P$($A$$cap$$B$) = $P(A)P(B)$ to prove independence

If I take $n=3$: I get the set $({1,2,3})$ that can be arranged in $3$ ways that satisfy the

condition of "$a_2$ $gt$ $a_1$". This is out of the total $8$ choices.

So, $Pr(B)$ $=$ $frac{3}{8}$

For $Pr(A)$, I don't know what they mean by this. "$a_n$ = n". Isn't $a_n$ always going to be $n$? So, $Pr(A)$ = 1?

For $P$($A$$cap$$B$) = $frac{3}{8}$ $.$ 1 = $frac{3}{8}$

$P$($A$$cap$$B$) = $P(A)P(B)$ = $frac{3}{8}$ = Independent!

Question:

b) Let $n$ $ge$ $5$ be an integer. Consider a uniformly random permutation $a_1$,$a_2$...$a_n$ of the set ${(1,2,...,n})$. Define the events:

A = "$a_1$ = 1"

B = "$a_n$ $=$ $5$"

What is $Pr$($A$$cup$$B$)?

a) $frac{1}{n}$ $-$ $frac{1}{n(n-1)}$

b) $frac{2}{n}$ $-$ $frac{1}{n(n-1)}$ (Answer)

c) $frac{2}{n}$ $-$ $frac{1}{n^2}$

Attempt:

For $n=5$, I can find the $P(A)$, $P(B)$ to be $frac{1}{5}$

I have to show $P$($A$$cup$$B$) = $P(A) $ + $ P(B)$ $-$ $P$($A$$cap$$B$).

I am struggling to determine $P$($A$$cap$$B$) for this. Would it just be $frac{1}{5}$ $.$ $frac{1}{5}$? That doesn't give me the correct solution for $n=5$ which should be 0.35.

(a)

If you have $3$ numbers and you permute them, you have only $3!=6$ possibilities.

$$Pr(B)=frac{1}{2}$$ from symmetry.

$$Pr(A)=frac1{n!}$$

since there is only one sequence that satisfies that condition.

Also $$Pr(A cap B)=Pr(A) ne Pr(A)Pr(B)$$

hence they are not independent.

(b)

$$Pr(A)=Pr(B)=frac{(n-1)!}{n!}=frac1n$$

$$Pr(Acap B)= frac{(n-2)!}{n!}=frac1{n(n-1)}$$

The idea is while two positions are fixed, we are free to permute the rest.

Try to avoid the temptation of thinking that $P(Acap B)=P(A)P(B)$ unless you can justify that they are independent.