St at ist ic s 270 - Lec t ure 5
Last class: measures of spread and box-plots Last Day - Began Chapter 2 on probability. Section 2.1 These Notes more Chapter 2 Section 2.2 and 2.3 Assignment 2: 2.8, 2.12, 2.18, 2.24, 2.30, 2.36, 2.40 Due: Friday, January 27 Suggested problems: 2.26, 2.28, 2.39
robabilit y robability of an event is the long-term proportion of times the event would occur if the experiment is repeated many times Read page 59-60 on Interpreting probability
robabilit y robability of event, A is denoted (A Axioms of robability: For any event, A, (S = 1 (A 0 If A 1, A 2,, A k are mutually exclusive events, These imply that 0 ( A 1
Disc ret e Uniform Dist ribut ion Sample space has k possible outcomes S= { e 1,e 2,,e k } Each outcome is equally likely (e i = If A is a collection of distinct outcomes from S, (A=
Ex am ple A coin is tossed 1 time S= robability of observing a heads or tails is
Ex am ple A coin is tossed 2 times S= What is the probability of getting either two heads or two tails? What is the probability of getting either one heads or two heads?
Ex am ple Inherited characteristics are transmitted from one generation to the next by genes Genes occur in pairs and offspring receive one from each parent Experiment was conducted to verify this idea ure red flower crossed with a pure white flower gives Two of these hybrids are crossed. Outcomes: robability of each outcome
Not e Sometimes, not all outcomes are equally likely (e.g., fixed die Recall, probability of an event is long-term proportion of times the event occurs when the experiment is performed repeatedly NOTE: robability refers to experiments or processes, not individuals
robabilit y Rules Have looked at computing probability for events How to compute probability for multiple events? Example: 65% of SFU Business School rofessors read the Wall Street Journal, 55% read the Vancouver Sun and 45% read both. A randomly selected rofessor is asked what newspaper they read. What is the probability the rofessor reads one of the 2 papers?
Addition Rules: If two events are mutually exclusive: Complement Rule ( ( ( ( B A B A B A ( ( ( B A B A ' ( 1 ( A A ( ( ( ( ( ( ( ( C B A C B C A B A C B A C B A
Example: 65% of SFU Business School rofessors read the Wall Street Journal, 55% read the Vancouver Sun and 45% read both. A randomly selected rofessor is asked what newspaper they read. What is the probability the rofessor reads one of the 2 papers?
Count ing and Com binat oric s In the equally likely case, computing probabilities involves counting the number of outcomes in an event This can be hard really Combinatorics is a branch of mathematics which develops efficient counting methods These methods are often useful for computing probabilites
Com binat oric s Consider the rhyme As I was going to St. Ives I met a man with seven wives Every wife had seven sacks Every sack had seven cats Every cat had seven kits Kits, cats, sacks and wives How many were going to St. Ives? Answer:
Ex am ple In three tosses of a coin, how many outcomes are there?
roduc t Rule Let an experiment E be comprised of smaller experiments E 1,E 2,,E k, where E i has n i outcomes The number of outcome sequences in E is (n 1 n 2 n 3 n k Example (St. Ives re-visited
Ex am ple In a certain state, automobile license plates list three letters (A-Z followed by four digits (0-9 How many possible license plates are there?
Tree Diagram Can help visualize the possible outcomes Constructed by listing the posbilites for E 1 and connecting these separately to each possiblility for E 2, and so on
Ex am ple In three tosses of a coin, how many outcomes are there?
Ex am ple - erm uat at ion Suppose have a standard deck of 52 playing cards (4 suits, with 13 cards per suit Suppose you are going to draw 5 cards, one at a time, with replacement (with replacement means you look at the card and put it back in the deck How many sequences can we observe
erm ut at ions In previous examples, the sample space for E i does not depend on the outcome from the previous step or sub-experiment The multiplication principle applies only if the number of outcomes for E i is the same for each outcome of E i-1 That is, the outcomes might change change depending on the previous step, but the number of outcomes remains the same
erm ut at ions When selecting object, one at a time, from a group of N objects, the number of possible sequences is: The is called the number of permutations of n things taken k at a time Sometimes denoted k,n Can be viewed as number of ways to select k things from n objects where the order matters
erm ut at ions The number of ordered sequences of k objects taken from a set of n distinct objects (I.e., number of permutations is: k,n = n(n-1 (n-k+ 1
Ex am ple Suppose have a standard deck of 52 playing cards (4 suits, with 13 cards per suit Suppose you are going to draw 5 cards, one at a time, without replacement How many permutations can we observe
Com binat ions If one is not concerned with the order in which things occur, then a set of k objects from a set with n objects is called a combination Example Suppose have 6 people, 3 of whom are to be selected at random for a committee The order in which they are selected is not important How many distinct committees are there?
Com binat ions The number of distinct combinations of k objects selected from n objects is: n n! k ( n k! n! C k, n n choose k Note: n!= n(n-1(n-2 1 Note: 0!= 1 Can be viewed as number of ways to select mthings taken k at a time where the order does not matter
Com binat ions Example Suppose have 6 people, 3 of whom are to be selected at random for a committee The order in which they are selected is not important How many distinct committees are there?
Ex am ple A committee of size three is to be selected from a group of 4 Conservatives, 3 Liberals and 2 NDs How many committees have a member from each group? What is the probability that there is a member from each group on the committee?
This document was created with Win2DF available at http://www.daneprairie.com. The unregistered version of Win2DF is for evaluation or non-commercial use only.