What can we say about probability?◦ It is a measure of likelihood, uncertainty,

possibility, …◦ And it is a number, numeric measure

Set: ◦ A set is a collection of distinct objects, considered

as an object in its own right◦ Example:

A class of stat225 students NBA teams Result of a football game for a team (win, lose, tie) Outcomes of rolling a die ( 1, 2, 3, 4, 5, 6)

Experiment:◦ A process that generates well-defined

outcomes.◦ *** You must know what could possibly happen

before performing the experiment.◦ Example:

Roll a die: YES ? Throw a stone out of window: NO.

Sample space◦ Don’t get confused with sample and sampling in

statistics. ◦ A sample space for an experiment is the set of

ALL experimental outcomes.◦ Example:

Toss a coin: {Head, Tail} Take an exam or quiz: { All the possible grades } Inspection of a manufactured part: {defective, non-

defective}

Some notes on sample space:1. It could be finite or infinite, i.e., there could be infinitely many possibilities, but you still must know all of them.

Example, choosing a point on a segment, there are infinitely many choices but you know it must be on the segment.2. Not all the sample points in the sample space are equally likely.

Example: Getting a royal flush, full house or just a pair in a poker game.

Event:◦ A collection of sample points.

It is an outcome from an experiment that may include one or more sample points.

◦ Examples: Toss a coin and get a head. Roll a die and get a 5. Toss a coin twice and get two heads. Roll a die and get an even number.

Complement of an event A:◦ Notation: Ac

◦ Also an event◦ Includes ALL sample points that are not in A.

Example: A: Roll a die and get an even number, Ac : Roll a die and get an odd number.

We use P(A) to represent the probability that one event occurs.

Clearly: P(A) + P(Ac)=1. A and Ac are called mutually exclusive,

which means any sample point call fall in either A only or Ac only, but not both.

Some of tossed a coin twice.◦ 1. What is the sample space:

{ HH, HT, TH, TT }◦ 2. Let A = { toss the coin twice and get two

heads}, then P(A)=?◦ 3. Let B = { toss the coin twice and get at least

one head}, then P(B)=?◦ 4. What is Bc , and P(Bc )=?

We have 20 fruits in a box, 10 apples, 6 pears and 4 peaches.◦ If A={pick a fruit from the box and get an apple},

then P(A)=?◦ If B={pick a fruit from the box and it is NOT an

apple}, then P(B)=?

50 balls are put in one box, 25 white, 15 red and 10 green.◦ A={ pick a ball and it is green }, P{A}=?

◦ B={pick a ball and it is colored}, P{B}=?

Venn diagram:◦ Illustrates the concept of complement.◦ Somewhat like a crosstabulation.

# 1. A class of 30 students took 2 midterms during a semester. 22 of them passed the first one and 26 of them passed the second one. If 3 students failed both, find the number of students who passed the each of the two midterms and failed the other.

Answer: 21 passed both, 1 passed the first but failed the second, 5 passed the second but failed the first

#2. We have a box and 30 balls, 16 are white and the rest are colored; 9 are plastic and the others are made of rubber. If there are 11 white rubber balls, show the breakdown of balls by color and material.

Answer: 5 white plastic; 11 white rubber; 4 colored plastic and 10 colored rubber.

#3. In a class of 30 students, 20 are male, 15 are white and 5 are black females. Assuming only no students of other race in the class. Find the break down of students by race and gender.

Answer: 10 black male; 5 black female; 5 white female and 10 white male.

In #1, what is the probability of finding a student who passed at least one midterm?◦ 27/30

In #2, pick a ball at random, what is the probability of getting a white plastic?◦ 5/30

In #3, pick a student from the class and what is the probability of getting a white female?◦ 5/30