Sample Space and Events Section 2.1

An experiment: is any action, process or phenomenon whose outcome is subject to uncertainty.

An outcome: is a result of an experiment.

Each run of an experiment results in only one outcome!

A sample space: is the set of all possible outcomes, S, of an experiment.

An event: is a subset of the sample space. An event occurs when one of the outcomes that belong to it occurs.

A simple (or elementary) event: is a subset of the sample space that has only one outcome.

Axioms, interpretation and properties Section 2.2

Probability: is a measure of the chance that an event might occur (before it really occurs). It is a function that uses events as input and results in a real number between 0 and 1 as output.

1) For any event A, Axioms (rules) of probability:

2) Probability of observing the sure event, S, if you run the experiment is:

3) If A1, A2, A3, …is an infinite collection of disjoint events (i.e. for any i and j where ), then

Axioms, interpretation and properties Section 2.2

Some other properties:1) For any event A:

P(A)+P(A’) = P(S) = 1 => P(A) = 1 – P(A’) 2) The probability of the empty set is:

3) For any events A and B where :

4) For any event A:

5) For any events A and B:

Axioms, interpretation and properties Section 2.2

Some other properties:Example:3) Forecasting the weather for each of the next three days

on the Palouse:

Experiment:

Set of possible outcomes (sample space), S:

Observing if the weather is rainy (R) or not (N) in each of those days.

{NNN, RNN, NRN, NNR, RRN, RNR, NRR, RRR}

Goal: Interested in whether you should bring an umbrella or not.

Axioms, interpretation and properties Section 2.2

Some other properties:Example:

Let A be the event that day 3 is rainy, P(A) = 0.4

Let B be the event that day two is rainy, P(B) = 0.7

Let C be the event days 2 and 3 are rainy, P(C) = 0.3

Find probability that either day 2 or 3 will be rainy.

3) Forecasting the weather for each of the next three days on the Palouse:

ANNR, RNR

S

BNRN,RRN

CNRR,RRR

Axioms, interpretation and properties Section 2.2

Some other properties:

Let A be the event that day three is rainy and P(A) = 0.4

Let B be the event that day two is rainy and P(B) = 0.7

Let C be the only days 2 and 3 are rainy and P(C) = 0.3

Find probability that either day 2 or 3 will be rainy.

= 0.4 + 0.7 – 0.3 = 0.8

Example:3) Forecasting the weather for each of the next three days

on the Palouse:

Axioms, interpretation and properties Section 2.2

Some other properties:Example:

Let A be the event that day 3 is rainy, P(A) = 0.4

Let B be the event that day two is rainy, P(B) = 0.7

Let C be the event days 2 and 3 are rainy, P(C) = 0.3

3) Forecasting the weather for each of the next three days on the Palouse:

Find probability that day 3 and not day 2 is rainy.

Axioms, interpretation and properties Section 2.2

Some other properties:Example:

Let A be the event that day 3 is rainy, P(A) = 0.4

Let B be the event that day two is rainy, P(B) = 0.7

Let C be the event days 2 and 3 are rainy, P(C) = 0.3

3) Forecasting the weather for each of the next three days on the Palouse:

Find probability that day 3 is not rainy.

Still in Ch2:

2.3 Counting Techniques

Counting Techniques Section 2.2

Counting methods: are used to count the simple events that comprise (partition) a sample spaces (a finite sample space usually). (Other purposes do exist!)

When these simple events are equally likely (like the die example) we can use these methods to calculate the probability of any event A by:

1) Finding the number of simple events (# of outcomes) that make up the event of interest N(A)

2) Finding the number of simple events (# of outcomes) that partition the sample space N

3) Calculating the probability P(A) = N(A)/N

Counting Techniques Section 2.2

Three counting techniques:

1) The multiplication rule

2) Permutations

3) Combinations

Counting Techniques Section 2.2

1) The multiplication ruleWe use the multiplication rule when each simple event (basically, each outcome) is made up of pairs (3-tuples, …, or k-tuples) and when order matters.

If a simple event is made up of k ordered tuples (k ordered elements) and if n1 is the number of possible choices for element 1; n2, number of possible choices for element 2; …; nk, number of possible choices for element k. Then the total number of possible simple events (and outcomes) is:

Counting Techniques Section 2.2

1) The multiplication rule

Example 1: Rolling two dice where (1,2) is not the same as (2,1); i.e. order matters. How many different outcomes (hence, simple events) can we observe if we ran this experiment?

Set of possible outcomes (sample space), S:

die1/die2 1 2 3 4 5 61 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Example 2: A college sports planning committee consists of 10 faculty members, 6 administrators, 8 undergraduate students and 3 graduate students. We want to form a subcommittee of 4 individuals to be in charge of planning basketball games for this year. How many different committees can we form given that we need to have a representative of each of the groups comprising the original committee? We first choose from professors, then administrators, then undergraduates, and finally graduates.

Counting Techniques Section 2.2

1) The multiplication rule

Counting Techniques Section 2.2

1) The multiplication rule

We also use the multiplication rule when each simple event is a result of sampling n distinct objects with replacement and when order matters.

Counting Techniques Section 2.2

1) The multiplication rule

Example 3: In how many ways can we obtain 5 cards with replacement from a deck of cards.

2) Permutations

Counting Techniques Section 2.2

We use the permutation rule when each simple event is a result of sampling n distinct objects without replacement and when order matters. (A special case of the multiplication rule that we have a name for.)

In this case a simple event is made up of k ordered tuples with n1= n, n2= n – 1 , …, and nk= n – k + 1. The total number of possible simple events is:

Counting Techniques Section 2.2

Example 4: In how many different ways can we arrange 4 letters from the alphabet if we are not allowed to use any letter more than one time? I care about order.

2) Permutations

Example 5: In how many ways can we obtain 5 cards without replacement from a deck of cards keeping track of the order of these cards.

Counting Techniques Section 2.2

2) Permutations

3) Combinations

Counting Techniques Section 2.2

We use the combination rule when each simple event is a result of sampling n distinct objects without replacement and when order does not matters.

In this case a simple event is made up of k-tuples with n1= n, n2= n – 1 , …, and nk= n – k + 1. The total number of possible simple events is:

3) Combinations

Counting Techniques Section 2.2

Example 6: In how many ways can we obtain 5 cards without replacement from a deck of cards regardless of their order?

Counting Techniques Section 2.2

Counting Techniques Section 2.2

Example 1: What is the chance that at least two individuals will have the same birthday in this class. Assume that a year has 365 days, and that each individual has an equal chance of having her/his birthday on any day of the year. Twins are not allowed in this class!

Example 8: Build the probability distribution that describes the chance of observing the sum of faces of two dice when rolled. Remember that:

Counting Techniques Section 2.2

die1/die2 1 2 3 4 5 61 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

And that each one of these outcomes is equally likely.