Statistics 270 - Lecture 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

Probability

• Probability 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

Probability

• Probability of event, A is denoted P(A)

• Axioms of Probability:• For any event, A, • P(S) = 1

• If A1, A2, …, Ak are mutually exclusive events,

• These imply that 1)(0 AP

0)( AP

Discrete Uniform Distribution

• Sample space has k possible outcomes S={e1,e2,…,ek}

• Each outcome is equally likely

• P(ei)=

• If A is a collection of distinct outcomes from S, P(A)=

Example

• A coin is tossed 1 time

• S=

• Probability of observing a heads or tails is

Example

• 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?

Example

• 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

• Pure red flower crossed with a pure white flower gives

• Two of these hybrids are crossed. Outcomes:

• Probability of each outcome

Note

• 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: Probability refers to experiments or processes, not individuals

Probability Rules

• Have looked at computing probability for events

• How to compute probability for multiple events?

• Example: 65% of SFU Business School Professors read the Wall Street Journal, 55% read the Vancouver Sun and 45% read both. A randomly selected Professor is asked what newspaper they read. What is the probability the Professor reads one of the 2 papers?

• Addition Rules:

• If two events are mutually exclusive:

• Complement Rule

)()()()( BAPBPAPBAP

)()()( BPAPBAP

)'(1)( APAP

)()()()()()()()( CBAPCBPCAPBAPCPBPAPCBAP

• Example: 65% of SFU Business School Professors read the Wall Street Journal, 55% read the Vancouver Sun and 45% read both. A randomly selected Professor is asked what newspaper they read. What is the probability the Professor reads one of the 2 papers?

Counting and Combinatorics

• 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

Combinatorics

• Consider the rhymeAs I was going to St. IvesI met a man with seven wivesEvery wife had seven sacksEvery sack had seven catsEvery cat had seven kitsKits, cats, sacks and wivesHow many were going to St. Ives?

• Answer:

Example

• In three tosses of a coin, how many outcomes are there?

Product Rule

• Let an experiment E be comprised of smaller experiments E1,E2,…,Ek, where Ei has ni outcomes

• The number of outcome sequences in E is (n1 n2 n3 …nk )

• Example (St. Ives re-visited)

Example

• 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 E1 and connecting these separately to each possiblility for E2, and so on

Example

• In three tosses of a coin, how many outcomes are there?

Example - Permuatation

• 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

Permutations

• In previous examples, the sample space for Ei does not depend on the outcome from the previous step or sub-experiment

• The multiplication principle applies only if the number of outcomes for Ei is the same for each outcome of Ei-1

• That is, the outcomes might change change depending on the previous step, but the number of outcomes remains the same

Permutations

• 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 Pk,n

• Can be viewed as number of ways to select k things from n objects where the order matters

Permutations

• The number of ordered sequences of k objects taken from a set of n distinct objects (I.e., number of permutations) is:

• Pk,n=n(n-1) … (n-k+1)

Example

• 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

Combinations

• 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?

Combinations

• The number of distinct combinations of k objects selected from n objects is:

• “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

nkCnkn

n

k

n,!)!(

!

Combinations

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?

Example

• A committee of size three is to be selected from a group of 4 Conservatives, 3 Liberals and 2 NDPs

• How many committees have a member from each group?

• What is the probability that there is a member from each group on the committee?