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.

## Randomly unfair coin.

onsider a machine that makes unfair coins. A coin that comes out of the machine has probability of head when tossed. The itself is a random variable sampled from uniform distribution (on [0,1]). We have some coin with unknown . We toss the coin and obtain head. What is the distribution of the result of the next toss of the same coin?

Let denote (head) at -th toss. We are interested in . We introduce a uniform mesh on and calculate using the formula ( Bayes formula ): According to the formula ( Total probability rule ), Hence,

 Notation. Index. Contents.