Chapter 4 Elementary Probability Theory

What is Probability?

Probability is a numerical measure between 0 and 1 that describes the likelihood that an event will occur. Probabilities closer to 1 indicate that the event is more likely to occur. Probabilities closer to 0 indicated that the event is less likely to occur.

Note:

P(A) = probability of event A; you read it as “P of A”.

P(A)=1, the event A is certain to occur

P(A)=0, the event A is certain to not occur

Binary number works like this…1 means it’s true, 0 means false.

See if you understand this statement:

“There are only 10 types of people in the world: those who understand binary, and those who don't”

Anyways…Probability Assignments

1) A probability assignment based on intuition incorporates past experience, judgment, or opinion to estimate the likelihood of an event

2) A probability assignment based on relative frequency uses the formula Probability of event=relative frequency=

Where f is the frequency of the event occurrence in a sample of n observations

3) A probability assignment based on equally likely outcomes uses the formula Probability of event=

Examples

Intuition – NBA announcer claims that Kobe makes 84% of his free throws. Based on this, he will have a high chance of making his next free throw.

Relative frequency – Auto Fix claims that the probability of Toyota breaking down is .10 based on a sample of 500 Toyota of which 50 broke down.

Equally likely outcome - You figure that if you guess on a SAT test, the probability of getting it right is .20

Group Work

Create a situation for each of the probability assignments. (intuition, relative frequency, equally likely outcome)

Show me

Law of Large Numbers

In the long run, as the sample size increases, the relative frequencies of outcomes get closer to the theoretical (or actual) probability value

Example: The more numbers you ask, the more likelihood that P(getting a girl’s real number)=1

Law of Large Numbers examples:

The more numbers you ask, the more likelihood that P(getting a (hot) girl’s real number)=1

Then after collecting all the numbers, the more girls you ask out on a date, the more likelihood that P(getting a date)=1

Some other real life examples:

Casino (the more you play, the more you lose)

Insurance (the more people you insure, the less the likelihood the company have to pay for the insurance benefits)

Statistical Experiment

Statistical experiment or statistical observation can be thought of as any random activity that results in a definite outcome

An event is a collection of one or more outcomes of a statistical experiment or observation

Simple event is one particular outcome of a statistical experiment

The set of all simple events constitutes the sample space of an experiment

Example: Blue eyes vs Brown eyes (relating to biology)

Brown eyes’ genotype is Bb or BB

Blue eyes’ genotype is bb

If your Dad has Brown eyes (and his dad has blue eyes) and your Mom has blue eyes, what’s the probability that you have blue eyes?

Answer (using sample space)

B b

b Bb bb

b Bb bb

Dad

Mom

P(blue eyes)=

Group Work (use sample space):

You are running out of time in a true/false quiz. You only have 4 questions left! How should you guess?

P(all false)= P(3 false)=

P(all true)= P(2 false)=

P(1 true)= P(1 false)=

P(2 true)=

P(3 true)=

Answer

Your sample space should have 16 different combinations

P(all false)= 1/16 P(3 false)= 4/16

P(all true)= 1/16 P(2 false)= 6/16

P(1 true)= 4/16 P(1 false)= 4/16

P(2 true)= 6/16

P(3 true)= 4/16

You will probably choose 2 true and 2 false

TTTT FTTT TFTT TTFT

TTTF FFTT FTFT FTTF

TFFT TFTF TTFF FFFT

TFFF FTFF FFTF FFFF

Note:

The sum of the probabilities of all simple events in a sample space must equal 1

The complement of event A is the event that A does not occur. designates the complement of event A. Furthermore,

1) P(A)+P(= 1

P(event A does not occur)=P

Example:

P(getting A in Mr. Liu’s class)+P(not getting A in Mr. Liu’s class) =1

P(getting A in Mr. Liu’s class)=.15

What’s the P(not getting A in Mr. Liu’s class)?

Answer

P(not getting A in Mr. Liu’s class)= .85

Group Work

P(having a date on a Friday)=1/7

What’s the P(not having a date on a Friday)?

Answer

6/7

Homework Practice:

Pg 130 #1-6 (all), 7-13 (odd)

Compound Events

Consider these two situation

P(5 on 1st die and 5 on 2nd die)

P(ace on 1st card and ace on 2nd card)

What is the difference between these two situation?

The answer

In the first situation, the first result does not effect the outcome of the 2nd result.

In the second situation, the first result does effect the outcome of the 2nd result.

Independent

Two events are independent if the occurrence or nonoccurrence of one does not change the probability that the other will occur

What does it mean if two events are dependent?

Multiplication rule for independent events

P(A and B)=

This means event A AND event B both have to happen!!! You multiply the events. You find the probability of two events happening together.

This is the formula if event A and event B are independent.

What if the events are dependent?

Then we must take into account the changes in the probability of one event caused by the occurrence of the other event.

Sample Space

A B

A and B

General multiplication rule for any events

P(A and B)=

Or

P(A and B)=

What is P?

It is known as conditional probability = “Probability of event A given event B”

Quick group work: What is P?

Conditional Probability Example:

Your friend has 2 children. You learned that she has a boy named Rick. What is the probability that Rick’s sibling is a boy?

Take a guess

Answer

If you guessed ½ or 50%, that is incorrect.

First: Think about all the possible outcomes S {BB, BG, GB, GG}

What is P(boy and boy)?

What is P(boy)?

You want to find =

Group Work

A machine produce parts that’s either good (90%), slightly defective (2%) or obliviously broken (8%). The parts gets through an automatic inspection machine that is able to find the oblivious broken parts and throw them away. What is the probability of the quality part that make it through and get shipped?

Answer

P(Good given not broken)=

Relax!

Conditional Probability can be very intriguing and complicated. We won’t go into any more in depth…..or maybe….

Note:

Very important to understand about probability is that are the events dependent or independent.

Group Work

Suppose you are going to throw 2 fair dice. What is the probability of getting a 3 on each die?

A) Is this situation independent or dependent?

B) Create all the sample space (all the potential outcomes)

C) What is the probability?

Answer

A) Independent because one event does not affect the second event

B) You should have 36 total outcomes

C) 1/36

Group Work

I took a die away. Now you only have ONE die! Again you toss the die twice. What is the probability of getting a 1 on the first and 4 on the second try?

Answer

It is still an independent event!

1/36

Note:

The last two examples are considered multiplication rule, independent events.

Group Work

Mr. Liu has a 80% probability of teaching statistics next year. Mr. Riley has a 15% probability of teaching statistics next year. What is the probability that both Mr. Liu and Mr. Riley teach statistics next year?

Answer

.8*.15=.12 or 12% probability

Now comes the dependent events

Suppose you have 100 Iphones. The defective rate of iphone is 10%. What is the probability that you choose two iphones and both are defective?

Answer

P(1st defective camera)=10/100

P(2nd defective camera)=9/99

P(1st defective camera and 2nd defective camera)=

Group work

What is the probability of getting tail and getting a 3 on a die and getting an ace in a deck of cards?

Answer

P(tail)=1/2

P(3)=1/6

P(ace)=4/52=1/13

Addition Rules

You use addition when you want to consider the possibility of one event OR another occurring

Example:

P(Jack or King)=P(Jack)+P(King)=

Group Work: And or Or?

1) Satisfying the humanities requirement by taking a course in the history of Japan or by taking a course in classical literature

2) Buying new tires and aligning the tires

3) Getting an A in math but also in biology

4) Having at least one of these pets: cat, dog, bird, rabbit

Answer

1) or

2) and

3) and

4) or

Note:

Two events are mutually exclusive or disjoint if they cannot occur together. In particular, events A and B are mutually exclusive if P(A and B)=0

Addition rule for mutually exclusive events A and B

P(A or B)=P(A)+P(B)

General Rule for any events A and B

P(A or B)=P(A)+P(B)-P(A and B)

Remember in mutually exclusive events P(A and B)=0

Group Example:

Employee type

Democrat (D)

Republican (R)

Independent (I)

Row Total

Executive (E) 5 34 9 48

Production Worker (PW)

63 21 8 92

Column Total 68 55 17 140 grand total

a) Compute P(D) and P(E)b) Compute c) Are events D and E independent?d) Compute P(D and E)e) Compute P(D or E)

Answer

A) P(D)=68/140 P(E)=48/140

B)

C) Determine if P(D)=, they are not the same! So they are not independent

D) P(D and E)=5/140

E) P(D or E)=

Homework Practice:

Pg 146 #1,2,5,7,9,10,14,19

Conditional Probability extension

Bayes’s theorem: It uses conditional probabilities to adjust calculations so that we can accommodate new relevant information.

The special case where event B is partitioned into only two mutually exclusive events.

Formula

Example (Real Life Extension): How accurate is “one” pregnancy test?

Supposedly pregnancy test strip claims it is 99% accurate.

false-positive and false-negative

  NOT Pregnant Pregnant!!!!

Pregnancy Test Negative True Negative False Negative

Pregnancy Test Positive False Positive True

Positive!!

Procedure

So you want to know

=.50

so 26%

But if you do the test more often, then the accuracy of the test increases.

Trees and Counting Techniques

Tree diagram:

A tree diagram shows all the possible outcomes of an event.

All possible outcomes of an event are shown by a tree diagram.

Example using tree diagrams:

If a coin and a dice are tossed simultaneously, what is the probability of getting tail and even number?

Answer

1/4

Group Work using tree diagram:

You are on a sports team. What is the probability that out of three games, you win two of them?

Tree Diagram with Probability:

You have 7 balls, 4 are blue and 3 are green. What is the probability that when you pick the balls, you get green on 1st and blue one 2nd?

Shown in class

Group Work:

You make free throws 85% of the time. What is the probability of making at least one out of the three?

P(make 1 out of 3)=99.66% of the time

Factorials

! Means factorial

0!=1

1!=1

n!=(n)(n-1)(n-2)(n-3)….

What is 6! ?

6!=6*5*4*3*2*1=720

Combination vs Permutation

Combination:

Permutation:

Combination:

Order does not matter! It is not important If you have 3,1,2, it is the same as 1,3,2 because they all

have 1,2,3

Different Arrangement of things

Combination is choosing

Permutation:

Order does matter! It is important

Note: Permutation is position

Combination vs Permutation continue:

Both of them break down into two different category:

Combination with repetition (ex: ice cream scoops)

Combination without repetition (ex: lottery)

Permutation with repetition (ex: lock in locker room)

Permutation without repetition (ex: marathon race)

Reading activity:

http://www.mathsisfun.com/combinatorics/combinations-permutations.html

Read the different examples

Whew that was a lot of reading…now do some examples.

You have 8 people, what are the number of possible ordered seating arrangement for 5 chairs

Answer

Permutation,

Group Work:

Gamestop has 25 new games this month and you decided to buy 5 of them. How many different arrange of game you can have?

Answer

Combination=

Group Work: MEGA Million

What is the chance of winning the jackpot for MEGA Million?

You have 5 slots + 1 slot for MEGA number

The first 5 slots are numbers between 1-75, mega number is number between 1-15

Answer

.00000000386

Or .000000386%

1 in 258890850

Group Work: Powerball lottery

What is the chance of winning the Jackpot for Powerball?

You have 5 slots+1 slot for Power number

The first 5 slots are numbers between 1-56, powerball slot is number between 1-35

Answer

1 in 175,223,510

Interesting fact

Even though it’s harder to win the Jackpot, for overall winning chance, you have more chance for MEGA million than Powerball

Homework Practice

Pg 160 #1-27 every other odd