1 / 93

GAMES OF CHANCE

GAMES OF CHANCE. COUNTING TECHNIQUES. Methods to determine how many subsets can be obtained from a set of objects are called counting techniques. FUNDAMENTAL THEOREM OF COUNTING

guild
Télécharger la présentation

GAMES OF CHANCE

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. GAMES OF CHANCE

  2. COUNTING TECHNIQUES • Methods to determine how many subsets can be obtained from a set of objects are called counting techniques. FUNDAMENTAL THEOREM OF COUNTING If a job consists of k separate tasks, the i-th of which can be done in ni ways, i=1,2,…,k, then the entire job can be done in n1xn2x…xnk ways.

  3. Cities 1 2 . . . 12 A R B A R B A R B EXAMPLE • A travel agency offers weekend trips to 12 different cities by air, rail or bus. In how many different ways such a trip be arranged? Tree Diagram:

  4. THE FACTORIAL • Number of ways in which objects can be permuted. n! = n(n-1)(n-2)…2.1 0! = 1, 1! = 1 Example: Possible permutations of {1,2,3} are {1,2,3}, {1,3,2}, {3,1,2}, {2,1,3}, {2,3,1}, {3,2,1}. So, there are 3!=6 different permutations.

  5. COUNTING • Partition Rule: There exists a single set of N distinctly different elements which is partitioned into k sets; the first set containing n1 elements, …, the k-th set containing nk elements. The number of different partitions is

  6. COUNTING • Example: Let’s partition {1,2,3} into two sets; first with 1 element, second with 2 elements. • Solution: Partition 1: {1} {2,3} Partition 2: {2} {1,3} Partition 3: {3} {1,2} 3!/(1! 2!)=3 different partitions

  7. Example • How many different arrangements can be made of the letters “ISI”? 1st letter 2nd letter 3rd letter I I S S I S I I N=3, n1=2, n2=1; 3!/(2!1!)=3

  8. Example • How many different arrangements can be made of the letters “statistics”? • N=10, n1=3 s, n2=3 t, n3=1 a, n4=2 i, n5=1 c

  9. COUNTING (e.g. picking the first 3 winners of a competition) • Ordered, without replacement • Ordered, with replacement 3. Unordered, without replacement 4. Unordered, with replacement (e.g. tossing a coin and observing a Head in the k th toss) (e.g. 6/49 lottery) (e.g. picking up red balls from an urn that has both red and green balls & putting them back)

  10. PERMUTATIONS • Any ordered sequence of r objects taken from a set of n distinct objects is called a permutation of size r of the objects.

  11. EXAMPLE • Consider the case where 5 of 7 students are to be seated in a row. In how many ways these students can be seated?

  12. COMBINATION • Given a set of n distinct objects, any unordered subset of size r of the objects is called a combination. Properties

  13. EXAMPLE • A carton of 12 rechargeable batteries contains one that is defective. In how many ways can an inspector choose 3 of the batteries and • Get the one that is defective? • Not get the one that is defective?

  14. COUNTING

  15. EXAMPLE • How many different ways can we arrange 3 books (A, B and C) in a shelf? • Order is important; without replacement • n=3, r=3; n!/(n-r)!=3!/0!=6, or

  16. EXAMPLE, cont. • How many different ways can we arrange 3 books (A, B and C) in a shelf? 1st book2nd book3rd book A B C C B B A C C A B C A B A

  17. EXAMPLE • Lotto games: Suppose that you pick 6 numbers out of 49 • What is the number of possible choices • If the order does not matter and no repetition is allowed? • If the order matters and no repetition is allowed?

  18. PROBABILITY AND BAYES THEOREM

  19. PROBABILITY POPULATION SAMPLE STATISTICAL INFERENCE

  20. PROBABILITY: A numerical value expressing the degree of uncertainty regarding the occurrence of an event. A measure of uncertainty. • STATISTICAL INFERENCE: The science of drawing inferences about the population based only on a part of the population, sample.

  21. PROBABILITY • CLASSICAL INTERPRETATION If a random experiment is repeated an infinite number of times, the relative frequency for any given outcome is the probability of this outcome. Probability of an event: Relative frequency of the occurrence of the event in the long run. • Example: Probability of observing a head in a fair coin toss is 0.5 (if coin is tossed long enough). • SUBJECTIVE INTERPRETATION The assignment of probabilities to event of interest is subjective • Example: I am guessing there is 50% chance of raining today.

  22. PROBABILITY • Random experiment • a random experiment is a process or course of action, whose outcome is uncertain. • Examples Experiment Outcomes • Flip a coin Heads and Tails • Record a statistics test marks Numbers between 0 and 100 • Measure the time to assemble Numbers from zero and abovea computer

  23. PROBABILITY • Performing the same random experiment repeatedly, may result in different outcomes, therefore, the best we can do is consider the probability of occurrence of a certain outcome. • To determine the probabilities, first we need to define and list the possible outcomes

  24. Sample Space • Determining the outcomes. • Build an exhaustive list of all possible outcomes. • Make sure the listed outcomes are mutually exclusive. • The set of all possible outcomes of an experiment is called a sample space and denoted byS.

  25. O1 O2 Sample Space:  = {O1,O2,…,Ok} Sample Space a sample space of a random experiment is a list of all possible outcomes of the experiment. The outcomes must be mutually exclusive and exhaustive. Simple Events The individual outcomes are called simple events. Simple events cannot be further decomposed into constituent outcomes. Event An event is any collection of one or more simple events Our objective is to determine P(A), the probability that event A will occur.

  26. Sample Space Uncountable (Continuous ) Countable Finite number of elements Infinite number of elements

  27. EXAMPLES • Countable sample space examples: • Tossing a coin experiment S : {Head, Tail} • Rolling a dice experiment S : {1, 2, 3, 4, 5, 6} • Determination of the sex of a newborn child S : {girl, boy} • Uncountable sample space examples: • Life time of a light bulb S : [0, ∞) • Closing daily prices of a stock S : [0, ∞)

  28. EXAMPLES • Examine 3 fuses in sequence and note the results of each experiment, then an outcome for the entire experiment is any sequence of N’s (non-defectives) and D’s (defectives) of length 3. Hence, the sample space is S : { NNN, NND, NDN, DNN, NDD, DND, DDN, DDD}

  29. Assigning Probabilities • Given a sample space S ={O1,O2,…,Ok}, the following characteristics for the probability P(Oi) of the simple event Oi must hold: • Probability of an event: The probability P(A), of event A is the sum of the probabilities assigned to the simple events contained in A.

  30. Assigning Probabilities • P(A) is the proportion of times the event A is observed.

  31. Intersection • The intersection of event A and B is the event that occurs when both A and B occur. • The intersection of events A and B is denoted by (A and B) or AB. • The joint probability of A and B is the probability of the intersection of A and B, which is denoted by P(A and B) or P(AB).

  32. Union • The union event of A and B is the event that occurs when either A or B or both occur. • At least one of the events occur. • It is denoted “A or B” OR AB

  33. Complement Rule • Thecomplement of event A(denoted by AC) is the event that occurs when event A does not occur. • The probability of the complement event is calculated by A and AC consist of all the simple events in the sample space. Therefore,P(A) + P(AC) = 1 P(AC) = 1 - P(A)

  34. MUTUALLY EXCLUSIVE EVENTS • Two events A and B are said to be mutually exclusive or disjoint, if A and B have no common outcomes. That is, A and B =  (empty set) • The events A1,A2,… are pairwise mutually exclusive (disjoint), if Ai  Aj =  for all i  j.

  35. EXAMPLE • The number of spots turning up when a six-sided dice is tossed is observed. Consider the following events. A: The number observed is at most 2. B: The number observed is an even number. C: The number 4 turns up.

  36. S 2 1 1 1 A A A 3 5 2 2 B B B C 4 4 6 6 4 6 AB 2 VENN DIAGRAM • A graphical representation of the sample space. AB AC = A and C are mutually exclusive

  37. AXIOMS OF PROBABILTY(KOLMOGOROV AXIOMS) Given a sample space S, the probability function is a function P that satisfies 1) For any event A, 0  P(A)  1. 2) P(S) = 1. 3) If A1, A2,… are pairwise disjoint, then

  38. THE CALCULUS OF PROBABILITIES • If P is a probability function and A is any set, then a. P()=0 b. P(A)  1 c. P(AC)=1  P(A)

  39. THE CALCULUS OF PROBABILITIES • If P is a probability function and A and B any sets, then • P(B  AC) = P(B)P(A  B) • If A  B, then P(A)  P(B) c. P(A  B)  P(A)+P(B)  1 (Bonferroni Inequality) d. (Boole’s Inequality)

  40. EQUALLY LIKELY OUTCOMES • The same probability is assigned to each simple event in the sample space, S. • Suppose that S={s1,…,sN} is a finite sample space. If all the outcomes are equally likely, then P({si})=1/N for every outcome si.

  41. Addition Rule For any two events A and B P(A  B) = P(A) + P(B) - P(A  B)

  42. ODDS • The odds of an event A is defined by • It tells us how much more likely to see the occurrence of event A. • P(A)=3/4P(AC)=1/4 P(A)/P(AC) = 3. That is, the odds is 3. It is 3 times more likely that A occurs as it is that it does not.

  43. Joint, Marginal, and Conditional Probability • We study methods to determine probabilities of events that result from combining other events in various ways. • There are several types of combinations and relationships between events: • Intersection of events • Union of events • Dependent and independent events • Complement event

  44. CONTINGENCY (CROSS-TABULATION) TABLES • Presents counts of two or more variables

  45. Joint, Marginal, and Conditional Probability • Joint probability is the probability that two events will occur simultaneously. • Marginal probability is the probability of the occurrence of the single event. The joint prob. of A2 and B1 The marginal probability of A1.

  46. Intersection • Example 1 • A potential investor examined the relationship between the performance of mutual funds and the school the fund manager earned his/her MBA. • The following table describes the joint probabilities.

  47. P(A1 and B1) Intersection • Example 1 – continued • The joint probability of [mutual fund outperform…] and […from a top 20 …] = .11 • The joint probability of[mutual fund outperform…] and […not from a top 20 …] = .06

  48. Intersection • Example 1 – continued • The joint probability of [mutual fund outperform…] and […from a top 20 …] = .11 • The joint probability of[mutual fund outperform…] and […not from a top 20 …] = .06 P(A1 and B1) P(A2 and B1)

  49. P(A1 and B1)+ P(A1 and B2) = P(A1) P(A2 and B1)+ P(A2 and B2) = P(A2) Marginal Probability • These probabilities are computed by adding across rows and down columns

  50. Marginal Probability • These probabilities are computed by adding across rows and down columns = + + =

More Related