Section 5.1 and 5.2Section 5.1 and 5.2ProbabilityProbability

Probability; it’s all chance!Probability; it’s all chance!

The big idea in probability is that The big idea in probability is that chance behavior is unpredictable chance behavior is unpredictable in the short run but follows a in the short run but follows a regular, predictable pattern in regular, predictable pattern in the long run.the long run.

Probability is a mathematical Probability is a mathematical model for what SHOULD model for what SHOULD happen!!happen!!

A Common QuestionA Common Question

The probability of tossing a coin and it The probability of tossing a coin and it landing on Heads is 0.5. Theoretically, then, landing on Heads is 0.5. Theoretically, then, if I toss the coin 10 times, I should get 5 if I toss the coin 10 times, I should get 5 Heads. However, with such a small number Heads. However, with such a small number of tosses there is a lot of room for variability.of tosses there is a lot of room for variability. There are two games involving flipping a coin.There are two games involving flipping a coin.

Game 1: You win $2 if you throw 40% - 60% heads.Game 1: You win $2 if you throw 40% - 60% heads. Game 2: You win $10 if you throw more than 75% Game 2: You win $10 if you throw more than 75%

heads.heads. For which game would you rather toss the coin 5 For which game would you rather toss the coin 5

times? 500 times?times? 500 times?

The Idea of ProbabilityThe Idea of Probability

Random does NOT mean haphazard. Random Random does NOT mean haphazard. Random events have an order that emerges in the long events have an order that emerges in the long run.run.

Random – individual outcomes are uncertain, Random – individual outcomes are uncertain, but there is a regular distribution of outcomes but there is a regular distribution of outcomes in a large number of repetitions.in a large number of repetitions.

Probability – proportion of times the outcomes Probability – proportion of times the outcomes would occur in a very long series of would occur in a very long series of repetitions.repetitions.

Independent – the outcome of one trial does Independent – the outcome of one trial does not influence the outcomes of any other trial not influence the outcomes of any other trial (ex. Rolling a die).(ex. Rolling a die).

PropertiesProperties

Remember that probabilities are like Remember that probabilities are like proportions; therefore, probabilities proportions; therefore, probabilities will always be a number between 0 will always be a number between 0 and 1 (inclusive).and 1 (inclusive).

The probability that I draw a 15 from The probability that I draw a 15 from a standard deck of cards is 0.a standard deck of cards is 0.

The probability that I draw a red or a The probability that I draw a red or a black card from a standard deck is 1.black card from a standard deck is 1.

Probability ModelsProbability Models

Sample spaceSample space SS is the set of all the possible is the set of all the possible outcomes.outcomes. If I toss 2 coins, and count the number of If I toss 2 coins, and count the number of

heads, the sample space is {0, 1, 2}.heads, the sample space is {0, 1, 2}. An An eventevent is any outcome or set of outcomes of a is any outcome or set of outcomes of a

random phenomenon. random phenomenon. An event for tossing 2 coins might be getting An event for tossing 2 coins might be getting

at least 1 head.at least 1 head. A A probability modelprobability model is a mathematical description is a mathematical description

– it lists the sample space and the probabilities – it lists the sample space and the probabilities associated with each event.associated with each event.

Word of the Day:Word of the Day:EnumerationEnumeration

It will be critical for you to be able to It will be critical for you to be able to list (enumerate) all the possible list (enumerate) all the possible outcomes of a random phenomenon.outcomes of a random phenomenon. Examples: List all the possible Examples: List all the possible

outcomes. How many are there? Then outcomes. How many are there? Then write the sample space. write the sample space.

Roll 2 dice Roll 2 dice Roll 1 die and toss a coinRoll 1 die and toss a coin

Two techniques can help us Two techniques can help us with enumeration…with enumeration…

Tree DiagramsTree Diagrams Draw the “branches” for the first task, and Draw the “branches” for the first task, and

then from each of those branches, draw the then from each of those branches, draw the branches for the second task.branches for the second task.

Let’s do this to represent the sample space for Let’s do this to represent the sample space for throwing one die and flipping a coin.throwing one die and flipping a coin.

Multiplication PrincipleMultiplication Principle Multiply the number of outcomes for the first Multiply the number of outcomes for the first

task by the number of outcomes for the second task by the number of outcomes for the second task. Do this for rolling two dice.task. Do this for rolling two dice.

ExampleExample

List the sample space for how many List the sample space for how many boys you could have within three boys you could have within three children.children.

Give the probability model for this Give the probability model for this situation.situation.

Probability RulesProbability Rules

Any probability is a number between (and Any probability is a number between (and including) 0 and 1.including) 0 and 1.

All possible outcomes together must have All possible outcomes together must have probability = 1.probability = 1.

The probability that an event does NOT occur The probability that an event does NOT occur is one minus the probability that the event is one minus the probability that the event does occur.does occur.

If two events have no outcomes in common, If two events have no outcomes in common, then probability that one OR the other occurs then probability that one OR the other occurs is the sum of their individual probabilities. is the sum of their individual probabilities.

Any probability is a number Any probability is a number between (and including) 0 and between (and including) 0 and

1.1.

In symbols:In symbols:

0 ≤ P(A) ≤ 10 ≤ P(A) ≤ 1

Probabilities can not be negative!Probabilities can not be negative! Probabilities can not be greater than Probabilities can not be greater than

1!1!

All possible outcomes together All possible outcomes together must have probability = 1.must have probability = 1.

In symbols.. The probability of the whole sample space: In symbols.. The probability of the whole sample space: P(S) = 1P(S) = 1

Sample question: A randomly selected student is asked to Sample question: A randomly selected student is asked to respond yes, no, or maybe to the following question, “Do you respond yes, no, or maybe to the following question, “Do you intend to vote in the next presidential election?” The sample intend to vote in the next presidential election?” The sample space is {yes, no, maybe}. Which of the following represents a space is {yes, no, maybe}. Which of the following represents a legitimate assignment of probabilities for this sample space?legitimate assignment of probabilities for this sample space? A) .4, .4, .2A) .4, .4, .2 B) .4, .6, .4B) .4, .6, .4 C) .3, .3, .3C) .3, .3, .3 D) .5, .3, -.2D) .5, .3, -.2 E) None of the aboveE) None of the above

The probability that an event does The probability that an event does NOT occur, called the NOT occur, called the

complement( ) is one minus the complement( ) is one minus the probability that the event does probability that the event does

occur.occur. In symbols: In symbols:

Sample question: If you choose a card Sample question: If you choose a card at random from a well-shuffled deck of at random from a well-shuffled deck of 52 cards, what is the probability that 52 cards, what is the probability that the card is not a heart?the card is not a heart?

( ) 1 ( )CP A P A

CA

P(Heart) = 13/52 = ¼

P(Not a Heart) = 1 – P(Heart) = 1 – ¼ = ¾

If two events have no outcomes in If two events have no outcomes in common, then probability that one OR common, then probability that one OR

the other occurs is the sum of their the other occurs is the sum of their individual probabilities.individual probabilities.

Having no outcomes in common is called Having no outcomes in common is called DISJOINT or MUTUALLY EXCLUSIVE.DISJOINT or MUTUALLY EXCLUSIVE.

This rule is very important!This rule is very important! It’s called the addition rule for disjoint It’s called the addition rule for disjoint

events.events. In symbols: In symbols: P(A or B) = P(A U B) = P(A) + P(B)P(A or B) = P(A U B) = P(A) + P(B)

This is read

“A union B.”

Sample QuestionSample Question

Which of the following pairs of events Which of the following pairs of events are disjoint?are disjoint? a) A: The odd numbers; B: the number 5a) A: The odd numbers; B: the number 5 b) A: the even numbers; B: the numbers b) A: the even numbers; B: the numbers

greater than 10greater than 10 c) A: the numbers less than 5; B: all c) A: the numbers less than 5; B: all

negative numbersnegative numbers d) A: the numbers above 100; B: all d) A: the numbers above 100; B: all

negative numbersnegative numbers e) A: negative numbers; B: odd numberse) A: negative numbers; B: odd numbers

More SymbolsMore Symbols

{A ∩ B} “A intersect B” – this means {A ∩ B} “A intersect B” – this means the outcomes that A and B have in the outcomes that A and B have in commoncommon

If A ∩ B = Ø, then A and B are If A ∩ B = Ø, then A and B are disjoint!disjoint!

What is A U AWhat is A U ACC?? What is A ∩ AWhat is A ∩ ACC??

WHY?1ØØ

Another thing to rememberAnother thing to remember

The probability that something will The probability that something will occur occur AT LEAST ONCEAT LEAST ONCE

1 – P(event does not happen)1 – P(event does not happen)how many attemptshow many attempts

Example: If the probability that I pass Example: If the probability that I pass any given Statistics test is 70% and I any given Statistics test is 70% and I have 10 tests this semester, what is the have 10 tests this semester, what is the probability that I pass at LEAST ONE?! probability that I pass at LEAST ONE?!

HomeworkHomework

Read Chapter 5 starting on pg Read Chapter 5 starting on pg 299; Chpt 5 #45, 50, 56299; Chpt 5 #45, 50, 56