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**Extra info for Gaussian Mixture Models and Probabilistic Decision**

**Example text**

This is essentially J. M. Keynes "Principle of Indifference" [18]. To use it in a concrete case, number five identical marbles from one to five, put them in a sack, mix them up, close your eyes, and choose one. The numbers have no influence on the draw since we cannot see them. Then what is the probability that marble one is drawn first? What is the probability that marble five is drawn last? What is the probability that both happen: marble one is first and marble five is last? Clearly, by symmetry all five marbles are equally likely to be the first choice, so the probability that number one is first is 1/5.

These are clearly not independent-A and B are identical-but they satisfy the equation. 26. A pair of fair dice is rolled. What is the conditional probability that one of the dice is a four, given that the sum is seven? 27. Three students are chosen at random from a class of ten girls and twelve boys. What is the probability that at least one is a boy, given that at least one is a girl? 28. A fair coin is tossed. If it is heads, one fair die is rolled. If it is tails, two fair dice are rolled. (a) What is the probability that the number of spots showing is 2?

This is not hard to see. The first row is 1, 1, which equals (), (1) and the second row, 1, 2, 1 is just (o), (1), (). For any n, (0n) = (n) = 1, which is where the 1's come from. 6. ) There is a congenial combinatorial interpretation of Pascal's triangle involving a stroll thru the triangle itself. 6) as a by-product. If we stroll thru the triangle, we follow a path. The path begins at the apex, and goes down one step at a time, each step going from a number to one of the two numbers directly below it.

### Gaussian Mixture Models and Probabilistic Decision

by Charles

4.3