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Understanding Probability and Independence in Probability Theory

Learn about sample spaces, random variables, events, probability functions, joint distributions, and combinatorics in the realm of probability and independence. Find out the best strategies for winning in casino games and dealing with counterplots.

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Understanding Probability and Independence in Probability Theory

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  1. Probability & Independence

  2. Sample space • Random variable • Probability

  3. Random variable

  4. Sample point ( ): the object where selection is made Sample space ( ) : the set of all sample points

  5. Random variable ( , , …. ): • a characteristic (number, color, etc.) of sample points. • a function converts sample points to their characteristics. • a function from sample space to the set of real numbers.

  6. Random variables

  7. Random variable generates subsets of the sample space. • Event: a subset of the sample space

  8. Each point in layout corresponds to an event. C A B E

  9. Probability is a function converts events to a number between 0 and 1.

  10. : sample space, the total set, the universal event : complementary event of : null event

  11. : the set of even numbers : multiples of 3 : multiples of 6 : multiples of 2 or 3

  12. 3 6 9 12 15 18 21 24 27 30 33 36 00 2 5 8 11 14 17 20 23 26 29 32 35 0 7 1 4 34 10 13 16 19 22 25 28 31

  13. 3 6 9 12 15 18 21 24 27 30 33 36 00 2 5 8 11 14 17 20 23 26 29 32 35 0 7 1 4 34 10 13 16 19 22 25 28 31

  14. : sample space, the total set, universal event, roulette, chocolate box, urn : set of subsets of layout

  15. Roulette Holes Characteristics of holes sample points, sample space random variables Layout set of subsets of sample space

  16. How to get money from casino ?

  17. Think coin tossing game. Bet $100 + “all the amount you lose” every time. Then you will win at least a time, then stop there.

  18. $500 House margin Using chips Betting limit

  19. “Who are willing to play a gambling game, seeing rising sun?”

  20. How to get money from casino ? The best strategy is the one that casinos want to keep out of. That is the strategy leaving casino as soon as possible.

  21. Counterplots of casino: High quality accommodation Far away location

  22. Joint distribution • Conditional distribution • Independence • Combinatorics

  23. Joint dist’n, Marginal dist’n

  24. Independence of random variables

  25. joint pdf, marginal pdf, independence

  26. conditional distribution f(x|y)

  27. X

  28. (0) (X)

  29. (X) (o) regardless of independence

  30. ?

  31. How many ways to give an order to n people ? a a b

  32. How many ways to give an order to r people selected from n people ? --- No order for (n-r) people. Give an full order to n people, and disregard the order of the last (n-r) people.

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