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 2Section 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

Discrete Uniform Distribution

- Sample space has k possible outcomes

Se1,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

- 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 hardreally
- Combinatorics is a branch of mathematics which

develops efficient counting methods - These methods are often useful for computing

probabilites

Combinatorics

- 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

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,nn(n-1) (n-k1)

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

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?