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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.

## Two headed coin drawn from a bin of fair coins.

bin holds 999 fair coins and one two-headed coin. A coin was drawn at random from the bin. The drawn coin was tossed ten times. We do not return it to the bin. Every time the result of the tossing is head. What is the probability that the drawn coin is the two-headed coin?

We introduce the random events These events have the properties We denote as the observed sample of ten tosses. We are interested in calculation of Prob . By the formula ( Inversion_remark ) we have Clearly, The Prob may be determined from the ( Total probability rule ) Therefore,

 Notation. Index. Contents.