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.

## Dealing a pair in the "hold' em" poker.

ou have been dealt a couple of cards from a well shuffled 52 card deck. What is the probability that you have a pair?

The index will identify a pair so that corresponds to a pair of two's, corresponds to a pair of 3's, and so fourth... aces correspond to . We have the n-union below is a total event: , hence, the n-union also consists of disjoint events: for , hence, we now apply the formula ( Total probability rule ) The value of every term above is independent of . Moreover, . Hence, because after there are 51 cards remaining and 3 of them are "two".

 Notation. Index. Contents.