You probability wonder what

we’re going to do next!

Probability Basics Experiment

an activity with observable results or outcomes

Sample space the set of all possible outcomes for an

experiment Event

any subset of the sample space

Probability Basics

P(E) = n(E)

n(S) (classical) or

f

n (experimental)

where P(E) means the probability of an event occurring, n(E) means the number of individual outcomes in the event, and n(S) means the number of individual outcomes in the sample space.

Flip a coin A well-known statistician named

Karl Pearson once flipped a coin 24,000 times and recorded _______ “heads”; this result was extremely close to the theoretical probability and expected number of heads.

P(H) = _____ E(H) = _________

Spinners

A

B C

D

Spin each spinner once. Find P(A).

A

B

C

Spinners If S = {1, 2, 3, 4, 5, . . ., 22, 23,

24}, find the probability of a: Prime number Even number Number less than 10 Number less than 3 or greater than 17 Number less than 12 and greater than

9

Dice Roll a single die once. Find the

following probabilities: P(number greater than 4 or less

than 2) P(odd or even) P(greater than 10) P(at least 3)

Odds

Odds for an event = P(for an event)/P(against the event)

If the odds for a horse winning a race are given, find the probability that the horse wins the race.

(a) 2:5 (b) 5:1 (c) 1:30

Experimental Probabilities

Conduct a poll to determine the probability: That a Gordon transfer student drives

a truck. That a Gordon student has enjoyed a

meal at the Dining Hall. That a Gordon student has visited the

BCM.

More vocabulary Complementary events

Everything else in the sample space Examples:

If A = rolling a 1 or a 2 on a die, “A complement” is rolling a 3, 4, 5, or 6

If R = it rains today, = it doesn’t rain today

R

Cards

Find the probability of drawing an ace from a standard deck of playing cards.

Find P(face card)

Find P(card with a value between 4 and 9)

More vocabulary Mutually exclusive (disjoint)

events When one event occurs, the other

cannot possibly occur Examples:

If A = even # on the roll of a die and B = 3 or 5,0 B) andP(A

P(A or B) Mutually exclusive events

Non-mutually exclusive events

P(A B) n A n B

n(S) = P(A) + P(B)

P A B n A n B n A B n S = P(A) + P(B) – P A B

P(A or B) Draw a card out of a standard 52-

card deck. Find the probability that the card is either black or an ace.

Roll a die once. If A = “even number on the die” and B = “rolling a 5 or 6”, find P(A or B).

Fundamental Counting Principle

If event M can occur in m ways and after it has occurred, event N can occur in n ways, then event M followed by event N can occur in m x n ways.(A tree diagram helps!)

Fundamental Counting Principle

How many outcomes are there for flipping 3 coins?

How many outcomes are there for rolling 2 dice? 3 dice?

If an ice cream shop has 32 flavors from which to choose and 7 toppings, how many different possibilities can I choose?

Fundamental Counting Principle

If automobile license plates consist of 4 letters followed by 3 digits, how many different license plates are possible if letters and digits may be repeated?

Multi-Stage Experiments

For any multi-stage experiment, the probability of the outcome along any path of a tree diagram is equal to the product of the probabilities along the path.

Problem

If the chance for success on the first stage of a rocket firing procedure is 96%, the second stage is 98%, and the final stage is 99%, find the probability of success on all three stages of the rocket firing procedure.

Flipping coins List the sample space for two

coins. (Use a tree diagram.) Find the probability of at least

one head. List the sample space for three

coins. (Use a tree diagram.) Find the probability of exactly

two heads.

Rolling Two Dice List the sample space. Find the probability of a 3 on the

first and a 3 on the second. Find the probability of a sum of 7. Find the probability of a sum of 10

or more. Find the probability that both

numbers are even.

Rolling Two Dice

Independent events

When the outcome of one event has no influence on the outcome of a second event, the events are independent.

For any independent events A and B,P(A and B) = P(A) x P(B).

Draw a ball from a container, replace it, and then draw a second ball.

Find the probability of a red, then a red.

Find P(no ball is red).

Find P(at least one red).

Find P(same color).

Dependent events

When the outcome of one event has an influence on the outcome of a second event, the events are dependent.

Draw a ball from a container, don’t replace it, and then draw a second ball.

Find the probability of a red, then a blue.

Find P(no ball is red).

Find P(same color).

Dependent vs. Independent events

Consider a bag that contains 219 coins of which 6 are rare Indian pennies. For the given pair of events A and B, complete parts (a) and (b) below.  A: When one of the coins is randomly selected, it is one of the Indian pennies.B: When another one of the coins is randomly selected, it is also one of the Indian pennies.a. Determine whether events A and B are independent or dependent. (If two events are technically dependent but can be treated as if they are independent according to the 5% guideline, consider them to be independent.)b. Find P(A and B), the probability that events A and B both occur.

A bag contains the letters of the word “probability”.

Draw 4 letters, one by one, from the bag. Find the probability of picking the letters of the word “baby” if the letters are drawn

With replacement

Without replacement

Consider the following 2 containers.

If a container above is selected at random, and then a letter is selected at random from the chosen container, what is the probability that the letter chosen is an M?

MATH HAT

#1 #2

For a challenge!

The Prisoner Problem

The Birthday Problem

Geometric Probabilities

If a dart hits the target below, find the probability that it hits somewhere in region 1.

1

2

3

4

21

2

The radius of the inner circle is 2 units and the radius of the outer circle is 4 units.

Games

A game is played by drawing 3 cards and replacing the card each time. A player wins if at least one face card is drawn. Find the probability of winning this game.

Find the probability of a “Yahtzee” in one throw of 5 dice.

Using Simulations

Flipping a coin

Rolling a die

Find the probability of a married couple having 2 baby girls.

Conditional Probabilities When the sample space of an

experiment is affected by additional info

AP

BAP A)|P(BA)given P(B

Conditional Probabilities If A = “getting a tail on the 1st toss of a

coin” and B = “getting a tail on all 3 tosses of a coin”, find P(B|A).

What is the probability of rolling a 6 on a fair die if you know you rolled an even number?

Conditional Probabilities With an auto insurance company, 60% of

its customers are considered low-risk, 30% are medium-risk, and 10% are high-risk. After a study, the company finds that during a 1-yr period, 1% of the low-risk drivers had an accident, 5% of the medium risk drivers had an accident, and 9% of the high risk drivers had an accident. If a driver is selected at random, find the probability that the driver will have had an accident during the year.

Factorial Notation Compute:

3!

7!5!2! 4!

6!

Permutations From n objects, choose r of them

and arrange them in a definite order. The number of ways this can be done is

! rn

!n Prn

Correspondences

How many differentways can 4 swimmers (Al, Betty, Carole, and Dan) be placed in 4 lanes for a swim meet?

Permutations

If there are 12 players on a little league baseball team, how many ways can the coach arrange batting orders, with 9 positions on the field and at bat?

Permutations

How many arrangements are possible with the letters in the word:

algebra?

statistics?

Combinations From n objects, choose subsets of

size r (order unimportant). The number of ways this can be done is

! rn !r

!n

!r

rPn Crn

Combinations

With 9 club members, how many different committees of 4 can be selected to attend a conference?

Braille Activity

Permutations & Combinations There are 10 players on a U-6 soccer

team, and the coach picks 5 starters. How many different groups of starters can the coach choose?

There are 10 members of a club. How many different “slates” could the membership elect as president, vice-president, and secretary/ treasurer (3 offices)?

Probability (with permutations/combinations) Given a class of 12 girls and 9 boys,

in how many ways can a committee of 5 be chosen?

in how many ways can a committee of 3 girls and 2 boys be chosen?

What is the probability that a committee of 5, chosen at random, consists of 3 girls and 2 boys?