91Ó°ÊÓ

The circumference of a circle or ellipse is its perimeter. That is, if the circle were expanded out and straightened out to a line segment, the circumference would be the arc length of the circle. The perimeter, in general, is the length of any closed figure's curve.

Shown that line passing through the center of the circle such that all the points are on one side of that line.

Let

${\mathit{P}}_1,…,{\mathit{P}}_n$denote the n points.

A: event contains all the points in some semicircle.

$A_i$: event for all the points lie in the semicircle beginning at the point $P_i$

For each i,

$A_i⊂A$

Thus,

${∪}_{i=1}^n{A}_i⊆A$

Now,

For reverse inclusion:

If A occurs,

Then

The semicircle can be rotated clockwise until it begins at $P_i$.

That shows

$A_i$and ${∪}_{i=1}^n{A}_i$ occurs.

Such that

$A⊆{∪}_{i=1}^n{A}_i$

Therefore,

Let

$P_1,…,P_n$denote the n points.

A : event contains all the points in some semicircle.

$A_i$: event for all the points lie in the semicircle beginning at the point $P_i$If $P_i=P_j$,

Then

$A_i$and $A_j$cannot be mutually exclusive.

And

$P_k$lies in the semicircle going clockwise for $180°$and starts at $P_i=P_j$.

In such case,

Both $A_i$and $A_j$occur.

On the other hand,

If $P_i≠P_j$,

Then

Both $A_i$and $A_j$cannot occur.

Also,

If $P_j$is in the semicircle going clockwise for $180°$and starts at$P_i$.

Then

$P_i$is in the semicircle going clockwise for $180°$and starts at $P_j$.

Thus,

$A_i{A}_j⊆\left(P_i=P_j\right)$

Furthermore,

$P\left(P_i=P_j\right)=0$

Such that

$P\left(A_i{A}_j\right)=0$

Thus,

We can say that

$A_i$ are almost mutually exclusive, but not quite.

Let

$P_1,…,P_n$denote the n points.

A : event contains all the points in some semicircle.

$A_i$: event for all the points lie in the semicircle beginning at the point $P_i$

From Part (b),

We have

$P\left(A_i{A}_j\right)=0$

That follows

$P\left(A_i{A}_j{A}_k\right)=0$

And

It is also true for all higher order intersections.

Then

We have

$P\left(A\right)⊆P\left(\underset{i=1}{\overset{n}{∪}}{A}_i\right)$

By the identity of inclusion-exclusion:

$P\left(A\right)=\underset{i=1}{\overset{n}{∑}}P\left(A_i\right)-\underset{i<j}{∑}P\left(A_i{A}_j\right)+\underset{ixj<k}{∑}P\left(A_i{A}_j{A}_k\right)-…$

Since all other terms are zero,

$P\left(A\right)=\underset{i=1}{\overset{n}{∑}}P\left(A_i\right)$

Now,

Fix

$i∈\{1,…,n\}$

And

A randomly chosen$P_i$.

For $j≠i$,

The probability that the randomly chosen point $P_j$is in the semicircle going clockwise for $180°$and starts at

$P_i$is $1/2$.

$i∈\{1,…,n\}$

And

A randomly chosen $P_i$.

For $j≠i$,

The probability that the randomly chosen point $P_j$is in the semicircle going clockwise for $180°$and starts at

$P_{i}is1/2$.

Thus,

$P\left(A\right)=\underset{i=1}{\overset{n}{∑}}P\left(A_i\right)=n(1/2{)}^{n-1}$

Thus,