Essential Statistics Chapter 4

Transcription

1 1 Essential Statistics Chapter 4 By Navidi and Monk Copyright 2016 Mark A. Thomas. All rights reserved.

2 2 Probability the probability of an event is a measure of how often a particular event will happen if an experiment (such as tossing a coin) is performed repeatedly a sample space contains all of the possible outcomes of a probability experiment an event is an outcome or collection of outcomes from a sample space with the law of large numbers, the more often an experiment is repeated, the better the approximation of the results will be to its actual probability

3 3 Probability if A denotes an event, then the probability of event A is denoted P(A) probability values range from 0 to 1, inclusive, so 0<=P(A)<=1 events can be impossible, with P(impossible) =? or events can be certain, with P(certain) =? if a sample space has equally likely outcomes, and event A has k outcomes, then P(A) = number of outcomes in A number of outcomes in sample space = k / n

4 4 Probability Unusual Events an unusual event is an event in which the probability of it occurring is small for our needs, if the probability of an event is less than 0.05, we will consider it unusual the empirical method merely mandates the experiment be conducted a large number of times, so that the actual outcome approximates the probability

5 5 Probability Addition Rule (4.2) a compound event is an event that is formed by combining two or more events P(A or B) = P(A occurs or B occurs or both occur) the general addition rule will use addition to summarize the or categories

6 6 Probability Addition Rule (4.2) the general addition rule states: P(A or B) = P(A) + P(B) P(A and B) use the general addition rule if given only probability numbers, not a table to use for use with NON-mutually exclusive events (see next slide)

7 7 Probability Mutual Exclusion events are mutually exclusive if it is impossible for both events to occur Venn diagrams are often used to visually display mutually exclusive events

8 8 Probability Addition Rule for MUTEX the general addition rule for mutual exclusion states: P(A or B) = P(A) + P(B) for use with mutually exclusive events

9 9 Probabilities - Complements if A is an event, the complement of A is the event that A does not occur this is sometimes denoted with a bar above A, or with a c in superscript, AA cc either A or AA cc must occur, both cannot occur P(A) + P (AA cc ) = 1, thus P(AA cc ) = 1 P(A)

10 10 Probabilities - Syntax at most n means not more than n, or <= n at least n means >= n

11 11 Sampling Without Replacement (4-3) avoids choosing any member of the population more than once sampling without replacement removes the selected item from the population the 2 nd selection is dependent upon the first (is not independent) e.g. selecting two items from data set a, b, c, we get (a, b), (a, c), (b, a), (b, c), (c, a), (c, b)

12 12 Sampling With Replacement sampling with replacement replaces the selected item from the population the 2 nd selection is from the entire population, since the 1 st item is replaced the 2 nd selection is not dependent upon the first (is independent) e.g. selecting two items from data set a, b, c, we get (a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)

13 13 Counting (4.4) Fundamental Principle of Counting if a sequence of operations is to be performed, the number of ways to perform the sequence is found by multiplying the numbers of ways to perform each operation that is, if a operation can be performed in m ways, and a second operation can be performed n ways, the total number to perform the sequence is m x n (or mn)

14 14 Permutations a permutation is the number of different ways a group of items can be ordered to find the different ways a group of n items can be ordered, we use a mathematical concept called a factorial, denoted n! n! = n x (n-1) x (n-2) x 2 x 1 so 5! = 5 x 4 x 3 x 2 x 1 = 120 by definition, 0! = 1

15 15 Permutations cont. d a permutation of r items chosen from n items is a specific ordering of the r items this is denoted npr in practice, npr = n!/(n r )! or more simply, multiply the first r things starting at n 5P3 = 5 x 4 x 3 = 60

16 16 Combinations in some cases, we don t care about the ordering, just the number of objects chosen these are called combinations, denoted by ncr that is the number of combinations of r objects chosen from a group of n objects in practice, ncr = n!/(r!(n r )!)