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Probability

Probability. MSIT3000 Lecture 3. Objectives. Probability & counting Key words: f/N, permutations, combinations. Introduction to Probability Theory The axioms of probability. Types of events and rules to help assign probabilities to them. Text: 3.1-3.5. Probability and intuition.

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Probability

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  1. Probability MSIT3000 Lecture 3

  2. Objectives • Probability & counting • Key words: f/N, permutations, combinations. • Introduction to Probability Theory • The axioms of probability. • Types of events and rules to help assign probabilities to them. Text: 3.1-3.5

  3. Probability and intuition • What is the probability of flipping a coin and getting heads? • What is the probability of rolling a six-sided die and getting a 1?

  4. The f/N rule • f = frequency • N = Number (of possible outcomes) • Therefore the estimated probability for rolling a 1 is 1/6. • In order to apply this generally we need more terminology.

  5. Terminology • Experiment [Def 3.1] • An experiment is an act or process of observation that leads to a single outcome that cannot be predicted with certainty. • Sample Point [Def 3.2] • A sample point is the most basic outcome of an experiment. • Sample Space [ Denoted: ; Def 3.3] • The sample space of an experiment is the collection of all its sample points. • Event [ Denoted: E; Def. 3.4 ] • An event is a specific collection of sample points. • Probability [ Denoted: P(E); Ch. 3 ] • A number between 0 and 1 assigned to events; meant to indicate the likelihood of the occurrence of an event.

  6. Possible outcomes of rolling 2 dice: Die 1 value, Die 2 value 1,1 2,1 3,1 4,1 5,1 6,1 1,2 2,2 3,2 4,2 5,2 6,2 1,3 2,3 3,3 4,3 5,3 6,3 1,4 2,4 3,4 4,4 5,4 6,4 1,5 2,5 3,5 4,5 5,5 6,5 1,6 2,6 3,6 4,6 5,6 6,6 Sample spaceExperiment: Two dice, one green & one red.

  7. Events and probabilities • What is the probability of rolling Green = 2 and Red = 5? • What is the probability of rolling one 2 and one 5? • What is the probability of rolling a sum of 7? In these examples we counted the number of outcomes consistent with the event and divided by the total number of possible outcomes.

  8. Applying the f/N rule • What is the probability of rolling a sum less than or equal to 4? • Answer: There are 6 cells in which the sum is 4 or less. There are 36 cells in the sample space. Therefore, f/N = 6/36 = 1/6

  9. New problem – large sample spaces • It can quickly become difficult or impossible to count all the possible outcomes explicitly. • For instance, how many poker hands are there in five-card stud? • How many possible flushes are there? • We need formulas to help us count.

  10. Counting rules • If you had to decide on dinner and had the following choices, how many ways could you make up the menu from the following choices? • Chicken or meat? • Wine or beer? • Rice or pasta? • That gives you 2 x 2 x 2 = 8 options. • We can generalize this to: • Choose 1 from n1, 1 from n2, ..., 1 from n(k). • Then there are n(1) x n(2) x ... x n(k) options.

  11. Counting rules –Permutations • When we care about the order we choose, different ways to order the same set is called a permutation. • For instance, how many permutations are there of the letters a, b, and c? • We could list all the possibilities. • Or we could note that after choosing one for the first position, there are only 2 left for the second and then only one left for the first position. Hence the number of permutations is 3 x 2 x 1 = 6.

  12. Permutations and notation • Generalizing to n letters (or n objects), we have • n(n-1)(n-2)···1 permutations. • We define n! = n(n-1)(n-2)···1 • and we read n! as “n factorial”. • Example: How many different ways can a group of 4 walk into a room? • Answer: 4 x 3 x 2 x 1 = 24

  13. More permutations • What if we have 4 letters to choose from, but only want 2? • First we have 4 options for the first choice, and then 3 for the next. Hence there are 4 x 3 = 12 possibilities. • But 4 x 3 = 4!/2! • This generalizes to: Read: The number of permutations of n distinct objects taken r at a time, for r=0,1,2,...,n.

  14. Counting rules –Combinations • What if we don’t care about the order our selections are made. • For instance, how many hands of poker are there in a 52 card deck without jokers? • If we care about the order the cards are selected, the answer is nPr = 52!/(52-5)!=311,875,200 • But each hand would then be counted 5!=120 times. • Hence the actual number of different hands is: • is nCr = nPr /r! = 2,598,960

  15. Summary

  16. Taking stock: • We have introduced probability and the f/N rule; • and we have introduced counting rules for: • permutations and combinations. Problems in class: What is the probability of being dealt on hand of 5 cards and getting: • a flush, • 4 of a kind, • royal straight flush ?

  17. So what is probability • ‘Intuitive’ definition • If you perform an experiment a large number of times the relative frequency of an event will approximate the probability of the event. • Less intuitively: • The probability of an event is any number assigned to a possible outcome that conforms to the rules of probability.

  18. The Rules of Probability • 1. All sample point probabilities must lie between 0 and 1, inclusive. • 2. The probabilities of all the sample points within a sample space must sum to 1.

  19. Visualizing Probability I:Venn Diagrams E C D A B

  20. Visualizing Probability II:Probability Table

  21. Specific types of events & probabilities: • Marginal (a.k.a. Unconditional) probability: • This is the probability that an event A will occur, regardless of what happens to B. • Union probability: • The probability that events A or B will occur. • Joint probability (a.k.a. Intersection): • The probability that events A and B will occur. • Complement of an event: • The complement of event A consists of every sample point that is not in A. I.e. If A does not occur, the complement of A must occur. • Denoted Ac in this text. Often ‘Not A’ is used [see table] • Note: P(A) + P(Ac) = 1 [this is useful because it is often easier to work with Ac than with A itself]

  22. How do we assign probability? • Subjectively • Dice – fair or not? • Objectively • Relative frequency • How does the f/N rule from last time fit in?

  23. Facts regarding the probability of an event (E) • Denoted: P(E) • P(E) must be greater than or equal to zero and less than or equal to one. • P(E)  [0,1] • The sum of all probabilities of non-overlapping events equals one.

  24. Mutually exclusive events A B

  25. Mutually exclusive events II • Events A and B are mutually exclusive if AB contains no sample points [AB = ] • Then: P(AB) = P(A) + P(B)

  26. The Additive Rule • For any two events: • P(AB) = P(A) + P(B) – P(AB) B A

  27. Conditional Probability • Given that one event has occurred, what is the probability that another will occur? • Example: given that you have gotten an A in statistics, what is the probability that you will get a 4.0 for the semester? • Or – given that you have gotten a C in statistics, what is the probability that you will get a 4.0 for the semester?

  28. Notation & Definition • P(A|B) = P(AB)/P(B) • The probability that A will occur is equal to the probability that A and B will occur, divided by the probability that B will occur. A B

  29. Conditional Probability in a table:

  30. P(A | B)

  31. Conclusion • Objectives addressed: • Probability & counting • Key words: f/N, permutations, combinations. • Introduction to Probability Theory • The axioms of probability. • Types of events and rules to help assign probabilities to them. • Illustrative problems: • Exam 1A #14-16; 19 • Text: 3.14

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