# math

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## The Monty Hall problem in plain English

The following is based on the book ‘Das Ziegenproblem‘ by Gero von Randow.

Setup:

• There are 3 doors.
• 1 door has a car behind it.
• 2 doors have a goat behind it.
• The content behind the doors is randomly chosen.
• There are 2 candidates, A and B.
• There is 1 moderator who knows which door has a car behind it.

Actions:

1. A and B are asked to choose a door and both choose the same door.
2. The moderator chooses one door which has a goat behind it.
3. A and B are asked if they would like to switch their choice and pick the remaining door.
4. A always stays with his choice, the door that has been initially chosen by both A and B.
5. B always changes her choice to the remaining third door.

Repeat the actions 999 times:

If you repeat the above list of actions 999 times, given the same setup, what will happen?

Candidate A always stays with his initial choice. Which means that he will on average win 1/3 of all games. He will win 1/3*999, 333 cars.

But who won the remaining 666 cars?

Given the setup of the game, the moderator has to choose a door with a goat behind it. Therefore the moderator does win 0 cars.

Candidate B, who always switched her choice, after the moderator picked a door with a goat behind it, must have won the remaining 666 cars (2/3*999)!

1 candidate and 100 doors:

Alter the above setup of the game in the following way

• There are 100 doors.
• 1 door has a car behind it.
• 99 doors have a goat behind it.
• There is 1 candidate, A.

Alter the above actions in the following way

• The moderator opens 98 doors with goats behind them.

Now let’s say the candidate picks door number 8. By rule of the game the moderator now has to open 98 of the remaining 99 doors behind which there is no car.

Afterwards there is only one door left besides door 8 that the candidate has chosen.

You would probably switch your choice to the remaining door now. If so, the same should be the case with only 3 doors!

Further explanation:

Your chance of picking the car with your initial choice is 1/3 but your chance of choosing a door with a goat behind it, at the beginning, is 2/3. Thus on average, 2/3 of times that you are playing this game you’ll pick a goat at first go. That also means that 2/3 of times that you are playing this game, and by definition pick a goat, the moderator will have to pick the only remaining goat. Because given the laws of the game the moderator knows where the car is and is only allowed to open a door with a goat in it.

What does that mean?

On average, at first go, you pick a goat 2/3 of the time and hence the moderator is forced to pick the remaining goat 2/3 of the time. That means 2/3 of the time there is no goat left, only the car is left behind the remaining door. Therefore 2/3 of the time the remaining door has the car. Which makes switching the winning strategy.

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## Number of combinations with repetition

There are n = 4 sorts of candy to choose from and you want to buy k = 10 candies. How many ways can you do it?

This is a problem of counting combinations (order does not matter) with repetition (you can choose multiple items from each category). Below we will translate this problem into a problem of counting combinations without repetition, which can be solved by using a better understood formula that is known as the “binomial coefficient“.

First let us represent the 10 possible candies by 10 symbols ‘C’ and divide them into 4 categories by placing a partition wall, represented by a ‘+’ sign, between each sort of candy to separate them from each other

CC+CCCC+C+CCC

Note that there are 10 symbols ‘C’ and 3 partition walls, represented by a ‘+’ sign. That is, there are n-1+k = 13, equivalently n+k-1, symbols. Further note that each of the 3 partition walls could be in 1 of 13 positions. In other words, to represent various choices of 10 candies from 4 categories, the positions of the partition walls could be rearranged by choosing n-1 = 3 of n+k-1 = 13 positions

C++CCC+CCCCCC

CCCCCCCCCC+++

We have now translated the original problem into choosing 3 of 13 available positions.

Note that each position can only be chosen once. Further, the order of the positions does not matter. Since choosing positions {1, 4, 12} does separate the same choice of candies as the set of positions {4, 12, 1}. Which means that we are now dealing with combinations without repetition.

Calculating combinations without repetition can be done using the formula that is known as the binomial coefficient

n!/k!(n-k)!

As justified above, to calculate combinations with repetition, simply replace n with n+k-1 and k with n-1,

(n+k-1)!/(n-1)!((n+k-1)-(n-1))!

In our example above this would be (4+10-1)!/(4-1)!((4+10-1)-(4-1))! = 13!/3!10!. Which is equivalent to

(n+k-1)!/k!(n-1)!

because (4+10-1)!/10!(4-1)! = 13!/10!3! = 13!/3!10!, which is the same result that we got above.

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## A Guide to Bayes’ Theorem – A few links

### Beginner ### miscellaneous A law of probability that describes the proper way to incorporate new evidence into prior probabilities to form an updated probability estimate. Bayesian rationality takes its name from this theorem, as it is regarded as the foundation of consistent rational reasoning under uncertainty. A.k.a. “Bayes’s Theorem” or “Bayes’s Rule”.

Eliezer Yudkowsky is on bloggingheads.tv with the statistician Andrew Gelman.

Several different points of fascination about Bayes…

When looking further, there is however a whole crowd on the blogs that seems to see more in Bayes’s theorem than a mere probability inversion…

Bayesian statistics is a system for describing epistemological uncertainty using the mathematical language of probability.
Bayesian probability is one of the most popular interpretations of the concept of probability.

Edwin T. Jaynes was one of the first people to realize that probability theory, as originated by Laplace, is a generalization of Aristotelian logic that reduces to deductive logic in the special case that our hypotheses are either true or false. This web site has been established to help promote this interpretation of probability theory by distributing articles, books and related material. As Ed Jaynes originated this interpretation of probability theory we have a large selection of his articles, as well as articles by a number of other people who use probability theory in this way…

Bayesian statistics is so closely linked with induction that one often hears it called “Bayesian induction.” What could be more inductive than taking a prior, gathering data, updating the prior with Bayes Law, and limiting to the true distribution of some parameter?

Gelman (of the popular statistics blog) and Shalizi point that, in practice, Bayesian statistics should actually be seen as Popper-style hypothesis-based deduction. The problem is intricately linked to the “taking a prior” above.

Or, how to recognize Bayes’ theorem when you meet one making small talk at a cocktail party.

Still, I’m sure Blogger won’t mind me using their resources instead. The basic idea is that there’s a distinction between true values x and measured values y. You start off with a prior probability distribution over the true values. You then have a likelihood function, which gives you the probability P(y|x) of measuring any value y given a hypothetical true value x.

In other words, What is so special about starting with a human-generated hypothesis? Bayesian methods suggest what I think is the right answer: To get from probabilistic evidence to the probability of something requires combining the evidence with a prior expectation, a “prior probability”, and human hypothesis generation enables this requirement to be ignored with considerable practical success.

Andrew Gelman recently responded to a commenter on the Yudkowsky/Gelman diavlog; the commenter complained that Bayesian statistics were too subjective and lacked rigor.  I shall explain why this is unbelievably ironic…

Maybe this kind of Bayesian method for “proving the null” could be used to achieve a better balance.

Bayesian brain is a term that is used to refer to the ability of the nervous system to operate in situations of uncertainty in a fashion that is close to the optimal prescribed by Bayesian statistics.

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P.S.

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