Probability question.

There are also cases where things seem improbable to us, but are actually quite probable. For example in a group of 23 people or more, there is a more than 50% chance of two people sharing a birthday.

Originally posted by twhitehead
There are also cases where things seem improbable to us, but are actually quite probable. For example in a group of 23 people or more, there is a more than 50% chance of two people sharing a birthday.

And there are also cases of events having zero probability and still they happen all the time.

Originally posted by mtthw
There's a very good essay on this by Ian Stewart and Jack Cohen. Unfortunately I can't find a reference to it. But I remember one part where one of them had a bet that "a coincidence will occur" while walking through an airport in Sweden. They then bumped into someone they knew and hadn't see for ages.

This seems to be pretty interesting. If you ever get to find this article then please post a link to it.

Originally posted by adam warlock
This seems to be pretty interesting. If you ever get to find this article then please post a link to it.

Found it:

http://www.newscientist.com/article/mg15721174.800-thats-amazing-isnt-it---why-is-intuition-worse-than-useless-when-it-comes-to-spotting-real-coincidences-jack-cohen-and-ian-stewart-investigate.html

You need a subscription to read the article though, I'm afraid.

Originally posted by adam warlock
And there are also cases of events having zero probability and still they happen all the time.

Example please.

Originally posted by FabianFnas
Example please.

Pick a random real number between 0 and 1.

(A bit artificial, but it demonstrates the principle)

Look up the mathematical concept of "almost surely"

Originally posted by FabianFnas
Example please.

We already discussed this.

The definition of the probability of an event rests on the notion of you having to integrate the probability density on a given interval. If you only integrate on a number the integral is trivially zero.

I'm, of course, assuming that the probability density function is well behaved.

Originally posted by mtthw
You need a subscription to read the article though, I'm afraid.

Damn!

Originally posted by mtthw
Pick a random real number between 0 and 1.

(A bit artificial, but it demonstrates the principle)

Look up the mathematical concept of "almost surely"

There a nice story by Borges about this. It revolves around a book in a library that contains an infinite number of pages. Hence, the chance of opening it on any page is infinitely small. Yet the book opens nonetheless.

Originally posted by adam warlock
We already discussed this.

The definition of the probability of an event rests on the notion of you having to integrate the probability density on a given interval. If you only integrate on a number the integral is trivially zero.

I'm, of course, assuming that the probability density function is well behaved.

Those delinquent, ne'er-do-well, probability functions: it's vanishingly unlikely anything good will come of them.

Originally posted by adam warlock
We already discussed this.

The definition of the probability of an event rests on the notion of you having to integrate the probability density on a given interval. If you only integrate on a number the integral is trivially zero.

I'm, of course, assuming that the probability density function is well behaved.

Yes, we've discussed this before. In order to understand what you mean, I ask again.

Please give me an example from real life.

Originally posted by FabianFnas
Yes, we've discussed this before. In order to understand what you mean, I ask again.

Please give me an example from real life.

Measure a photon. What was the probability it had the energy you measured it to have before you measured it?

Originally posted by KazetNagorra
Measure a photon. What was the probability it had the energy you measured it to have before you measured it?

You mean, you measure a photon twice and the probability that it has (exactly) the same energy both times is zero, and yet it happens?

Can it also be so that you measure the temperature of an object and find it to be (exactly) to be pi degrees. A probability of zero, yet it happens that an object has that temp when you heat it from zero to 5 degrees? Have I got it right?

Sorry, I construct another thought experiment as I don't know if a photon has a constant temp for a while, or not, as a quantum effect.

Originally posted by FabianFnas
Yes, we've discussed this before. In order to understand what you mean, I ask again.

Please give me an example from real life.

Any kind of event whose sample space is continuous.

I think that your problem in this question is that you may be too hanged up on the frequency definition of probability and aren't very used in thinking that probability is a function that arises from an integral. I may be wrong but I really think this is the problem here.

But anywhoo:

Choose any number on the real line.

The probability of choosing a rational number is 0, yet almost everybody chooses a rational number when confronted with this question.
Mind that I didn't say that the probability of choosing a particular number is 0. What I said is that the probability of choosing a whole set of numbers (and this set is infinite) is 0.
To make matters worse the probability of choosing an irrational number is 1 and yet just go around and ask this question to 100 people and see how many of them choose an irrational number as answer...

Originally posted by adam warlock
But anywhoo:

Choose any number on the real line.

The probability of choosing a rational number is 0, yet almost everybody chooses a rational number when confronted with this question.
Mind that I didn't say that the probability of choosing a particular number is 0. What I said is that the probability of choosing a whole set of numbers (and this s ...[text shortened]... k this question to 100 people and see how many of them choose an irrational number as answer...

I would counter that probability also depends on the concept of randomness. So your example seems counter intuitive because the choices in question are not random. Whether this makes you claims wrong I am not certain, but you probably are.