Question

Give an example of an event which has probability zero but the event is not an impossible event and an example of an event which has probability one but the event is not the entire event space.

Answer 1

@perl

Answer 2

consider a continuous distribution of scores of some exam

the probability of getting a score of 75 is 0 as the area under curve is 0
but the event is not an impossible event

Answer 3

How can marks be a continuous distribution?? It is discrete..

Answer 4

i thought scores/perentiles can be continuous hmm
anyways you get the idea right ? just pick some continuous distribution and consider probability of a single value (without interval)

Answer 5

yeah for any continuous distribution P(X=a)=0.. we have this theorem. Actually marks are given either in whole number or of the form n+1/2 form, obviously the irrational marks are not given, hence there is a gap in the distribution and hence discrete..

Answer 6

maybe consider heights of ppl in america or something more continuous

Answer 7

Hmmm.:) so probability of hitting the dart board at the centre is 0??

Answer 8

by center you mean a point and not area of finite disk right ?

Answer 9

because center of dartboards usually have finite radius

Answer 10

yeah... the dart board looks like a continuous distribution and obviously normal distribution I think... I dont know the answer, just guessing..

Answer 11

http://img6a.flixcart.com/image/board/h/b/2/wood-o-plast-dart-board-set-14-inch-400x400-imadffg9er8gx3hr.jpeg

Answer 12

center is not a point so the probability for hitting inside the red disk is not 0

Answer 13

Hmm then.. if we take the red disk as the event space ?

Answer 14

im guessing it wont work as the area is finite

Answer 15

Hmm then?? we are not yet arriving at a satisfactory answer...

Answer 16

consider continuous variables like temperature / heights/time etc

Answer 17

A more abstract scenario: picture an \(n\)-faced "die". Each face has a \(\dfrac{1}{n}\) probability of occurring since they are uniformly distributed. Rolling the die, you have a \(\dfrac{1}{n}\to0\) chance of getting a particular number as \(n\to\infty\) (and as \(n\to\infty\) the die becomes a ball, so the face becomes a point on the surface of the ball).

Answer 18

As for the other scenario, the only way I can see that happening is if you're given two events, say \(x=a\) and \(x=b\), and you're explicitly given some sort of probability distribution of the form
\[p(x)=\begin{cases}1&\text{for }x=a\\0&\text{for }x=b\end{cases}\]

Answer 19

*not necessarily two events, there's nothing stopping you from using a continuous distribution, but you'd have to have something of the form
\[f(x)=\begin{cases}1&\text{for }a<x<b\\0&\text{else}\end{cases}\]
where \(x\) falling between \(a\) and \(b\) is a distinct event.

Answer 20

@SithsAndGiggles I will have to go for now.. When I will return , I will see your examples....

Answer 21

@SithsAndGiggles I see the examples are theoretically possible and in practice , not possible.

Answer 22

Ah but you asked for an example, not a realistic one ;)

Answer 23

Lol.:P.@SithsAndGiggles