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 I. Basic math.
 1 Conditional probability.
 A. Definition of conditional probability.
 B. A bomb on a plane.
 C. Dealing a pair in the "hold' em" poker.
 D. Monty-Hall problem.
 E. Two headed coin drawn from a bin of fair coins.
 F. Randomly unfair coin.
 G. Recursive Bayesian calculation.
 H. Birthday problem.
 I. Backward induction.
 J. Conditional expectation. Filtration. Flow of information. Stopping time.
 2 Normal distribution.
 3 Brownian motion.
 4 Poisson process.
 5 Ito integral.
 6 Ito calculus.
 7 Change of measure.
 8 Girsanov's theorem.
 9 Forward Kolmogorov's equation.
 10 Backward Kolmogorov's equation.
 11 Optimal control, Bellman equation, Dynamic programming.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Recursive Bayesian calculation.

wo identically looking black boxes hold two balls each. One of the boxes holds two black balls and the other box holds one black and one white ball. We randomly fix our attention on one of the boxes. We repeatedly take out one of the balls at random and then put it back. Suppose we did it times and each time this happens to be a black ball. What is the probability of recovering the white ball at the -rd draw?

We introduce a random variable Denote as the conditional distribution of after the -th draw. The variable ( =white ball, =black ball) is the result of the -th draw.

Before the first draw we have the distribution Hence, by the formula ( Total probability rule ), We compute the via the formula ( Inversion_remark ) We repeat the procedure for the draw 2: We finally compute the probability for the third draw:

 Notation. Index. Contents.