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This 3 page Class Notes was uploaded by Heli Patel on Sunday June 19, 2016. The Class Notes belongs to 3339 at University of Houston taught by Prof. C Poliak in Summer 2016. Since its upload, it has received 8 views. For similar materials see Statistics for the Sciences in Math at University of Houston.
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Date Created: 06/19/16
Probability 3.13.7 ● Randomness We call a phenomenon random if individual outcomes are uncertain. Chance behaviors unpredictable in the short run but has a regular and predictable pattern in the long run. ● Probability ○ The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. ○ If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct. ■ Classical method is use when all the experimental outcomes are equally likely. If n experimental outcomes are possible, a probability of 1/n is assigned to each experimental outcome. Example: Drawing a card from a standard deck of 52 cards. Each card has a 1/52 probability o ■ Relative frequency method is used when assigning probabilities is appropriate when data are available to estimate the proportion of the time the experimental outcome will occur if the experiment is repeated a large number of times. That is for any outcome, A, probability of A is ■ Subjective method of assigning probability is most appropriate when one cannot realistically assume that the experimental outcomes are equally likely and when little relevant data are available. ● Definition ○ A set is a collection of objects. ○ The items that are in a set called elements. ○ The sample space of a random phenomenon is the set of all possible outcomes. Ω is used to denote sample space ○ Notation Description ○ a ∈ A The object a is an element of the set A. ○ A ⊆ B Set A is a subset of set B. That is every element in A is also in B. ○ A ⊂ B Set A is a proper subset of set B. That is every element that is is in A is also in set B and there is at least one element in set B that is no in set A. ○ A ∪ B A set of all elements that are in A or B. ○ A ∩ B A set of all elements that are in A and B. ○ Ω Called the universal set, all elements we are interested in. ○ ∼A The set of all elements that are in the universal set but not in set A. ○ S i Ei E1 ∪ E2 ∪ . . ., the union of multiple sets ○ T i Ei E1 ∩ E2 ∩ . . ., the intersection of multiple sets ● Permutations ○ It allows one to compute the number of outcomes when r objects are to be selected from a set of n objects where the order of selection is important. The number of permutations is given by Where n! = n(n − 1)(n − 2)· · ·(2)(1) Rocode for n!: factorial(n) ● Allowing Repeated Values ○ When we allow repeated values, The number of orderings of n objects taken r at a time, with repetition is n r ■ how many ways can you write 4 letters on a tag using each of the letters C O U G A R with repetition? ■ 6^4 = 1296 ● Combinations ○ Counts the number of experimental outcomes when the experiment involves selecting r objects from a (usually larger) set of n objects. The number of combinations of n objects taken r unordered at a time is ○ ● Several Objects AT Once ○ The number of permutations, P, of n objects taken n at a time with r objects alike, s of another kind alike, and t of another kind alike is ○ Example: How many different words (they do not have to be real words) can be formed from the letters in the word MISSISSIPPI? ■ n(M) = 1 n = 11 ■ n(I) = 4 ■ n(S) = 4 P= .. 11! .. = 34650 ■ n(P) = 2 1! 4! 4! 2! ● Objects Taken of Circular ○ The number of circular permutations of n objects is (n − 1)!. ○ Example: In how many ways can 12 people be seated around a circular table? ○ (121)! = 11! = 39,916,800 ● Basic Probability Rules ○ 1. 0 ≤ P(E) ≤ 1 for each event E. ○ 2. P(Ω) = 1 ○ 3. If E1, E2, . . . is a finite or infinite sequence of events such that Ei ∩ Ej = ∅ for i 6= j, then P( T i Ei) = P i P(Ei). If Ei ∩ Ej = ∅ for all i 6= j we say that the events E1, E2, . . . are pairwise disjoint. ○ 4. Complement Rule: P(E ∩ ∼ F) = P(E) − P(E ∩ F). In particular, P( ∼E) = 1 − P(E). ○ 5. P(∅) = 0 ○ 6. Addition Rule: P(E ∪ F) = P(E) + P(F) − P(E ∩ F). ○ 7. If E1 ⊆ E2 ⊆ . . . is an infinite sequence, then P( S i Ei) = limi→∞P(Ei). ○ 8. IF E1 ⊇ E2 ⊇ . . . is an infinite sequence, then P( T i Ei) = limi→∞P(Ei).
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