# Thread: Probability (point on circle)

1. ## Probability (point on circle)

I have a word problem that is vexing me; the language is getting in the way of my reasoning...

"Let's say we have a circle with a circumference of 10. A tack will be placed somewhere along the circumference of this circle. What is the likelihood it will be placed at the top half of the circumference? What is the likelihood it will placed at exactly the top of the circumference?"

Presumably, the answer to the first question is .5

But what does 'exactly at the top' mean? Couldn't there be an infinite number of places around the circumference for the tack (as no units of measurement are provided)? And how does one determine the top?

Is this a trick logic question or am I missing something?

2. Originally Posted by Quixotic
I have a word problem that is vexing me; the language is getting in the way of my reasoning...

"Let's say we have a circle with a circumference of 10. A tack will be placed somewhere along the circumference of this circle. What is the likelihood it will be placed at the top half of the circumference? What is the likelihood it will placed at exactly the top of the circumference?"

Presumably, the answer to the first question is .5

But what does 'exactly at the top' mean? Couldn't there be an infinite number of places around the circumference for the tack (as no units of measurement are provided)? And how does one determine the top?

Is this a trick logic question or am I missing something?
The answer to the second part is zero. Consider if it had instead had asked what is the probability of it is placed on an arc that subtends $x$ radians from the centre, then (assuming a uniform distribution on the circumference, which you should have specified) the probability is $x/(2\pi)$.

Now as $x \to 0$ this probability goes to zero.

(as a general rule with a continuous probability distribution the probability of an discrete value occuring is zero)

CB

3. Originally Posted by CaptainBlack

Now as $x \to 0$ this probability goes to zero.

(as a general rule with a continuous probability distribution the probability of an discrete value occuring is zero)

CB
I'm grateful for the answer, but I still do not understand the reasoning behind it. Incidentally, I quoted the question from my worksheet exactly. No additional information was provided (and this isn't from a statistics course!).

My new question is this: how does one know or 'prove' that the probability of a discrete value occuring is zero? This sounds like one of those general principles I need to remember, but I don't understand it.

4. Originally Posted by Quixotic
I'm grateful for the answer, but I still do not understand the reasoning behind it. Incidentally, I quoted the question from my worksheet exactly. No additional information was provided (and this isn't from a statistics course!).

My new question is this: how does one know or 'prove' that the probability of a discrete value occuring is zero? This sounds like one of those general principles I need to remember, but I don't understand it.
Well if it says no more than you report the only correct answer is that there is insufficient data.

Essentially for a continuous random variable what I posted is the proof, with a minor modification for distributions other than the uniform. Though if you wait long enough someone will be along with a measure theory proof.

CB