Coincidence and the birthday paradox, page 1

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I don't quite know how to answer. For one thing, I hardly ever concern myself with coincidences/synchronicities which occur in day-to-day life; I have my hands full with grading, tutorials, papers, that sort of boring stuff

For another, it is often--rather, always--the case that we cherry-pick what is "relevant" and what is not. In fact, even if we read all the history books... the history books are all cherry-picked! If we could store a sufficiently large and well-distributed set of statistics, we could get a more conclusive answer.

Here is a page on the coincidences/synchronicities surrounding the number of "elevens" which show up around the 9/11 incident (plus a deliciously sarcastic retort). In this case I'd venture that we have on the order of 10^7 pieces of information.

Now, the explanation for those who are lost. It's good that people are curious about it. Maybe it will get people to do more math? I don't know.

Say you have some objects, represented by Xs. You have a group of people. Each person is allowed to pick one X. More than one person can pick the same X. What are the chances that more than one person picks the same X?

For the sake of simplicity, let's have 5 Xs and 2 people.

X X X X X

We can list out all the ways which 2 people can pick from 5 Xs by representing them as sequences of numbers: 11, 12, 13, 14, 15, 21, 22, 23, etc. For example, 42 means the first person picks the 4th X and the second person picks the 2nd X.

You can figure out how many ways there are in total by listing all the ways (there are 25 in total if you list them all out) or calculating. Since there are 5 choices for each person and 2 people, there are 5^2 = 25 ways.

This formula works for any N Xs and S people. For example, if N = 3 and S = 3 there are 3^3 = 27 ways. Although 27 is quite small, one would rather not list out all the ways for N = 365 and S = 23. That means it is a good idea to make a formula. And we have.

Now here's the tricky part. How many ways are there in which more than one person picks the same X? This time we have to dive straight in to the formula. The trick is to exclude the ways in which all people pick different Xs.

For the case of 5 Xs and 2 people, the first person has 5 choices, the second person has 4. So we have 5 * 4 = 20. It looks like this listed out: 12, 13, 14, 15, 21, 23, 24, 25, etc. We exclude these ways from the total, so we have 25 - 20 = 5.

For the case of 5 Xs and 3 people, the first person has 5 choices, the second has 4, the third has 3. So we have 5 * 4 * 3 = 60. There are 5^3 = 125 ways in total, so excluding them we have 125 - 60 = 65.

This kind of product, for instance 10 * 9 * 8 * 7, where we start with a number and multiply and decrease over and over, can be expressed using factorial notation: 3! = 3 * 2 * 1, 4! = 4 * 3 * 2 * 1, and so on. Therefore, 10! / 6! = 10 * 9 * 8 * 7.

Similarly, for the case of N Xs and S people, we exclude N! / (N - S)! ways.

To put it all together:
There are N! / (N - S)! ways to exclude
There are N^S ways in total
After excluding, we get N^S - N! / (N - S)!
Dividing by the total number of ways, we get (N^S - N! / (N - S)!) / N^S
Simplifying to 1 - N! / ((N - S)! * N^S)

Which is the collision function stated in the OP.

The birthday paradox is just a "surprising" result obtained from plugging in N = 365 and S = 23 into the collision function.

As an aside... how do people expect to wrap their heads around the serious stuff like astrophysics, classical thermo- and electrodynamics, and quantum mechanics while being stumped by probability/combinatorics?
edit on 2-5-2012 by Tadeusz because: (no reason given)

Exactly the kind of thing I was looking for. Thanks.

post by Cecilofs

I am aware of what probability theory is and what it isn't. It isn't supposed to be perfect. It is a model. What does "the probability of an event" mean? It is a ratio between the size of a set and the size of a subset (hence its restriction to the range [0, 1]). Example: we model the outcomes of a fair 6-sided die using the set {1, 2, 3, 4, 5, 6}. We model the desired outcomes using a subset. Let's say we want a 4. Then we say {4}. The size of {4} is 1 and the size of {1, 2, 3, 4, 5, 6} is 6. We find that the probability of getting a 4 is 1/6.

Indeed, theory can be detached from real life. Probability theory is purely theoretical and thus only applicable before taking any samples. What actually happens doesn't matter; I could roll the die a million times and get a million 4s. But the probability of getting a 4 is still 1/6, because we have already decided on using {1, 2, 3, 4, 5, 6} and {4}.

It seems to be a backwards way of thinking, so why do we use it? It is often easier--and thus more practical--than analyzing lots of samples.

What's the chances that both their parents went to that movie, then concieved around the same time, then the babies were born at around the same time, then they both grew up, came to the same place and both volunteered for your experiment? I am no mathematician, but I expect the odds are astronomical.

The space of possibilities is enormous. Anything could have happened to disrupt this chain of events: one of the parents has a car accident on the way to the hospital, one of the babies is born with a heart defect, one of the kids is sent to work in a salt mine. Yet, you can still model this space using a set, with each element representing a sequence of events instead of a single event. Even with astronomical odds against your hypothetical being chosen, it can still be chosen.

Here is an analogy. Suppose you shuffle a standard deck of 52 cards using a machine with such power that each permutation is equally likely. You shuffle using the machine. Out come the cards, sorted by rank and suit. Would that surprise you? But it would have been equally likely to see some meaningless jumble of cards. The point is that we determine what is meaningful. Maybe the order Ace of Clubs, Seven of Diamonds, Jack of Diamonds, etc. is meaningful to me.

As for Synchronicity: it is a matter of belief, much like belief in God. It is a concept derived from a somewhat liberal interpretation of quantum entanglement. For the interested, here is an article from Ars Technica about the misuse of QE. A passage from the article states:

Entanglement is delicate, rare, and short-lived.

It follows that "macro-entanglement" is nigh-improbable, amounting to solving an astronomical number of simultaneous wave equations.

I do not believe in Synchronicity; on the other hand, probability theory has already been applied to great effect. And with this statement you have shown me that Synchronicity is indeed a matter of belief:

you haven't convinced me that Synchronicity doesn't exist

That was never my intention in the first place.

I have already chosen to interpret the universe one way, and someone else may have chosen to interpret it a different way.

Be aware that I have also chosen not to involve myself in any mudslinging, as tempting as it may be.

It's quite a popular thing on which people disagree. A related question is: do we have free will? These are questions with no answers. All the same, I would like to hear your opinion: what does it really mean when some things are not random? Does it mean that there is a higher power with some measure of control over the universe?

My opinion: hard determinism with probability. Every state has a well-defined set of next states to "choose" from, but we don't know which. Science is essentially materialistic and cynical; it seeks to prove this assumption wrong by building more and more intricate theoretical models of the universe. If we do achieve this, then we would pretty much have hit the jackpot--discovered the higher power behind the universe. Even if we never get there, then we would have at least advanced as a species.

While a scientist would love to see synchronicity in action, he would smother it with more theory and ask for more evidence

reply posted on 8-5-2012 @ 02:51 AM by OccamsRazor04

You are doing what everyone does, that which makes the birthday scenario seem counterintuitive. This is the question you asked.
What's the chances that both their parents went to that movie, then concieved around the same time, then the babies were born at around the same time, then they both grew up, came to the same place and both volunteered for your experiment? I am no mathematician, but I expect the odds are astronomical.

This would be akin to asking what are the odds someone in the group of 23 shares YOUR birthday.

This is what you should have asked.

What's the chances that two sets of parents went to the same movie, then concieved around the same time, then the babies were born at around the same time, then they both grew up, came to the same place and both volunteered for the same experiment?

The difference being that you are falsely creating a coincidence by limiting the starting point to only two sets of parents. In reality you need to start with ALL parents who saw that movie.

Couples tend to see romantic movies.
Having sex after is not uncommon.
Conceiving on the same night will mean the babies will be born at the same time.
The parents were from the same area so the children will grow up in that area.
You may have 20 children born as the result of that steamy movie, you just need any 2 of them to end up at the experment to create the illusion of coincidence.

What you did was create coincidence, when in reality there is mathematical probabilities at work on an exponential level. Just because only 2 children showed up for the experiment does not mean they were the only 2 born. Some events have very low odds, but in a world with billions of people they will happen.
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