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5.1 Basic Probability Ideas

5.1 Basic Probability Ideas. Definition: Experiment – obtaining a piece of data Definition: Outcome – result of an experiment Definition: Sample space – list of all possible outcomes of an experiment

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5.1 Basic Probability Ideas

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  1. 5.1 Basic Probability Ideas • Definition:Experiment – obtaining a piece of data • Definition: Outcome – result of an experiment • Definition: Sample space – list of all possible outcomes of an experiment • Definition: Event – collection of outcomes from an experiment (a simple event is a single outcome) • Definition: Equally likely sample space – sample space in which all outcomes are equally likely

  2. 5.1 Basic Probability Ideas • Definition: Probability – A number between 0 and 1 (inclusive) that indicates how likely an event is to occur • Definition: n(A) – number of simple events in A • Definition: Theoretical probability of A – p(A)= n(A)/n(S) = (# outcomes of A) ÷ (# outcomes in the equally likely sample space)

  3. 5.1 Basic Probability Ideas • Law of Large Numbers – as the number of experiments increases without bound, the proportion of a certain event approaches a theoretical probability • The Law of Large Numbers does not say: If you throw a coin and get heads 10 times, that your probability of getting tails increases. P(heads) stays at 1/2

  4. 5.1 Basic Probability Ideas • Empirical Probability – relative frequency of an event based on past experiencep(A) = (# times A has occurred) ÷ (# observations)

  5. 5.1 Basic Probability Ideas • Properties of Probabilities: • 0 ≤ p(A) ≤ 1 • p(a) = 0  A is impossible, the event will never happen • p(a) = 1  A is a certain event, the event must happen • Let a1, a2, a3,…. an be all events in a sample space, then p(a1) + p(a2) + … + p(an) = 1

  6. 5.2 Rules of Probability • Definition: Complement of an event A – The event that A does not occur denoted AC • Properties: • P(A) + p(AC) = 1 • P(AC) = 1 – p(A) • P(A) = 1 – p(AC) • Odds in favor of event A – n(A):n(AC) orn(A) ÷ n(AC)

  7. 5.2 Rules of Probability • Example: p(A) = 1/3 then p(AC) = 1 – p(A) = 2/3odds in favor of A = p(A):p(AC) = 1/3:2/3 = 1:2odds against A = p(AC):p(A) = 2/3:1/3 = 2:1 • Probabilities from odds in favor –odds in favor = s:f (successes to failures)p(A) = s/(s + f)p(AC) = f/(s + f)

  8. 5.2 Rules of Probability • Joint probability tables – displays possible outcomes and their likelyhood of occurrence • Example: Given the following table of data:

  9. 5.2 Rules of Probability • Probability table for example (total of 80 people in the sample):

  10. 5.2 Rules of Probability • Simple probability tree: B Boy branches root G Girl

  11. 5.2 Rules of Probability • Probability trees are useful when events do not have the same probability (there is no equally likely sample space) • Problem solutions involving trees can become long if many branches are to be calculated (similar to the brute force method in section 4.5)

  12. 5.3 Probabilities of Unions and Intersections • Definition: The union of two events A and B is the event that occurs if either A or B or both occur in a single experiment. The union of A and B is denoted A BExample: (rolling a die – getting an even number or a perfect square) 2 4 6 5 3 1

  13. 5.3 Probabilities of Unions and Intersections • Definition: The intersection of two events A and B is the event that occurs if both A and B occur in a single experiment. The intersection of A and B is denoted A BExample: (rolling a die – getting an even number and a perfect square) 2 4 6 5 3 1

  14. 5.3 Probabilities of Unions and Intersections • Definition: mutually exclusive or disjoint events – events for which A  B =  (where  represents an event with no elements) • If A and B are mutually exclusive, then: • P(A  B) = 0 • P(A B) = P(A) + P(B) • Union Principle of Probability: P(A  B) = P(A) + P(B) - P(A  B)

  15. 5.4 Conditional Probability and Independence • Definition: The conditional probability of A given B is the probability of A occurring given that B has already occurred – denoted P(AB)When outcomes are equally likely: n(AB) n(B) P(AB) = • Conditional Probability Formula (outcomes not necessarily equally likely) P(AB)P(B) P(AB) =

  16. 5.4 Conditional Probability and Independence • Multiplication Principal:P(A  B) = P(B)  P(AB) • Tree diagrams – useful for conditional probability because each section of a branch is a probability conditional by the previous branches

  17. 5.4 Conditional Probability and Independence • Independence: Two events A and B are said to be independent if the occurrence of A does not affect P(B) and vice versa.A & B are independent if:P(AB) = P(A) or P(BA) = P(B) • Multiplication Principle for Independent Events:A & B are independent events  P(A  B) = P(A)  P(B)

  18. 5.5 Bayes’ Formula • Bayes formula for 2 cases: P(A)  P(BA) P(AB) = P(A)  P(BA) + P(AC)  P(BAC)

  19. 5.5 Bayes’ Formula • Bayes formula for n disjoint events: P(Ai)  P(BAi) P(AiB) = P(A1)  P(BA1) + P(A2)  P(BA2) + … + P(An)  P(BAn)

  20. 5.6 Permutations and Combinations • Multiplication Principle – given a tree with the number of choices at each branch being m1, m2, m3, … mn, then the number of possible occurrences is: m1 m2 m3 …  mn

  21. 5.6 Permutations and Combinations • Permutations: The number of arrangements of r items from a set of n items.Note: Order matters. n! (n – r)! nPr =

  22. 5.6 Permutations and Combinations • Combinations: The number of subsets of r items from a set of n items.Note: Order does not matter. n! (n – r)!  r! nCr =

  23. 5.6 Permutations and Combinations- summary of counting formulas Without replacement With replacement (order matters) Order doesnot matter (subsets) Order matters (arrangements) Multiplicationprincipal Permutation Combination

  24. 5.7 Probability and Counting Formulas • Example: A bag contains 4 red marbles and 3 blue marbles. Find the probability of selecting: • Two red marbles • Two blue marbles • A red marble followed by a blue marble # ways to pick 2 red# ways to pick any 2 marbles = 4C2 7C2 = 6/21 = 2/7 P(2 red) =

  25. 5.7 Probability and Counting Formulas # ways to pick 2 blue# ways to pick any 2 marbles P(2 blue) = = 3C2 7C2 = 3/21 = 1/7 chance of picking red on first chance of picking blue on second P(red then blue) = = 4/7  3/6 = 2/7

  26. 5.7 Probability and Counting Formulas • Birthday Problem: Suppose there are n people in a room. Find the formula for the probability that at least two people have the same birthday.Note: P(at least two birthdays the same) = 1 – P(no two birthdays the same)# ways for n people to have birthdays = 365n# ways for for n birthdays without repeats = 365Pnanswer = 1 – (365Pn  365n)

  27. 5.8 Expected Value • Expected Value – for a given sample space with disjoint outcomes having probabilities p1, p2, p3, … pn and a value (winnings) of x1, x2, x3, … xn , then the expected value of the sample space is:x1 p1+ x2 p2+ x3 p3+…. xn pn • Definition: A game is said to be fair if the cost of participating equals the expected winnings. • Expected winnings < cost  unfair to participant • Expected winnings > cost  unfair to organizers

  28. 5.9 Binomial Experiments • Definition: Binomial Experiment • The same trial is repeated n times. • There are only 2 possible outcomes for each trial – success or failure • The trials are independent so the probability of success or failure is the same for each trial.

  29. 5.9 Binomial Experiments • Binomial Probabilities:P(x successes) = nCr px  (1-p)n-x with x = 0,1,2,…,nwhere n is the number of trials, p is the probability of success, and x is the number of successful trials • Expected value of a binomial experiment = n  pnote: The most likely outcome “tends to be close to” the expected value.

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