Authors: Amanda Hoezee, Brandi Stewart, Courtney Johnson Mission Statement: The purpose of this wiki page is to assist in the education of probability and its components to high school students. Relevant Common Core State Standards:

Conditional Probability and Rules of Probability

Understand independence and conditional probability and use them to interpret data (S-CP.1-5)

Use the rules of probability to compute probabilities of compound events in a uniform probability model (S-CP.6-9)

Using Probability to Make Decisions

Calculate expected values and use them to solve problems (S-MD.1-4)

Use probability to evaluate outcomes of decisions (S-MD.5-7)

What is probability? In common usage, the word "probability" is used to mean the chance that a particular event, or set of events, will occur on a scale from 0 to 1.1

Can be expressed as a percentage between 0% and 100%.

Basic Probability Terms:

S-CP.1. Describe events as subsets of a sample space using characteristics of the outcomes, or as unions, intersections, or complements of other events.

Sample space: the sample space for a given set of events is the set of all possible values the events may assume.

Example: the set of all possible outcomes of two coin tosses is: {(H,H); (H,T), (T,H), (T,T)}2

Union: ("or") the union of two sets A and B is the set containing members of A or B, or both A and B.

Pronounced "A union B"

Can be calclated as such:

Intersection: ("and") The intersection of two sets A and B is the set of elements common to A and B3

Pronounced "A intersection B"

Complement: the complement of a set A is all outcomes not in the outcomes of set A. The complement of an event A is denoted Ac and is read "A complement."4 It can be calculated as follows: P(Ac) = 1 - P(A) For example, a card is chosen at random from a deck of cards. What is the probability that a card chosen is not a face card?

A = event that card chosen is a face card

Ac = event that card chosen is not a face card

P(Ac) = 1 - P(A)

= 1 - (12/52)

= 40/52

= 10/13

Disjoint: two sets are said to be disjoint if their intersection is the empty set, or the two sets have no elements in common

Let's Visualize This!

Unions, intersections, and complements are easily shown visually by Venn Diagrams...

Independence S-CP.2.Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

(See Multiplication Rule for Independent Events below)

Example: An executive on a business trip must rent a car in each of two different cities. Let A denote the event that the executive is offered a free upgrade in the first city and B represents the analogous event for the second ciry. Suppose P(A)=0.2, P(B)=0.3, P(Ac)=0.8, and P(Bc)=0.7. What is the probability that the executive is offered a free upgrade in at least one of the two cities? *

P(at least one) = 1 - P(Ac and Bc) =1-[P(Ac)*P(Bc)] =1-(.8)(.7) =.44

S-CP.3.** Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

For any two events A and B with P(B) > 0, the conditional probability of A given hat B had occurred is defined by:
Independence is when the outcomes of two events A and B do not affect each other. Therefore, independent events do not have conditional probability because the events are not dependent on each other.

Further application....Bayes' Rule! By definition, Bayes' Rule states: P(Hk|A) = P(A|Hk) P(Hk) / ∑i P(A|Hi) P(Hi) where the sum is i = 1...n. Example: A doctor is called to see a sick child. The doctor has prior information that 90% of sick children have the flu, while 10% are sick with measles. F = event that child is sick with the fluM = even that child is sick with measlesA well-known symptom of measles is a rash (denoted by R), P(R|M) = 0.95. However, occasionally children with the flu also develop a rash so that P(R|F) = 0.08. Upon examining the child, the doctor finds a rash. What is the probability that the child has measles?"5

Rules of Probability S-CP.7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the uniform probability model. S-CP.8.Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

Addition Rule: If A and B are separate set of events the then the probability of either event occurring is P(A or B) = P(A) + P(B) - P(A and B)

If the events A and B are mutually exclusive, meaning that it is not possible for both sets of events to occur at the same time, then P(A and B)=0 and the addition rule becomes P(A or B) = P(A) + P(B)6

Example:Suppose a high school consists of 25% juniors, 15% seniors, and the remaining 60% is students of other grades. The relative frequency of students who are either juniors and seniors is 40%. We can add the relative frequencies of juniors and seniors because no student can be both junior and senior.7P(J or S) = 0.25 + 0.15= 0.40

Multiplication Rule for Independent Events: If A and B are independent events then, P(A and B) = P(A) * P(B)8

Multiplication Rule for Dependent Events: If A and B are dependent events, then P(A and B) = P(A) * P(B|A)9

Permutations and Combinations S-CP.9.Use permutations and combinations to compute probabilities of compound events and solve problems.

Combinations:An unordered subset of k objects, taken from a set of n distinct objects is a combination of size k***

"n choose k"

Example: S = {a,b,c,d}

If k = 3, possible combinations are (a,b,c), (a,b,d), (a,c,d), (b,c,d)

Notation:

Example: If you have a deck of cards without the jokers, how many 5-card hands can you get?

= 2,598,960

Permutations: any ordered sequence of k objects taken from a set of n distinct objects***

Notation:

Example: A club of 5 people can elect one member as president and a different member as treasurer. How many different ways can members be elected? ***

Expected Value S-MD.1.Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. S-MD.2. Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. S-MD.3.Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

In order to understand the calculation of expected values, it's necessary to understand the concepts of random variables and the probability distribution function.

Random variable: any rule that associates a number with an outcome in the sample space***

Allows us to relate experimental outcomes to a numerical function of the outcomes

Often abbreviated by r.v.

Can be categotized as discrete or continuous

Example: Three cars are selected at random and each is categorized as having a diesel (D) or nondiesel (F) engine. If X (the random variable ) = the number of cars among the three with diesel engines, list each outcome in S and its associated X value.

Probability distribution function: describes probabilities of random variables occurring

For a discrete random variable, defined by p(x)= P(X=x)

Can be presented in multiple ways, such as:

Example: An automobile facility specializing in engine tune-ups knows that 45% of all tune-ups are done on four-cylinder automobiles, 40% on six-cylinder automobiles, and 15% on eight-cylinder automobiles. Let X = the number of cylinders on the next car to be tuned. The pdf is show above.***

Using the chart, what is P(X=6)?

P(X=6) = 0.40

S-MD.5. Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

Expected value: the weighted average of possible values of a random, where the weights are the probabilities10

Calculated by sum of each possible outcome multiplied by the probability of that particular outcome

Example: An individual who has car insurance from a certain company is randomly selected. Let X be the number of moving violations for which the individual was cited during the last 3 years. The pdf of X is given below.***

(a) Compute E(X)

(b) Suppose an individual with X violations incurs a surcharge of $100X2. Calculate the expected amount of the surcharge.

Probability

: Amanda Hoezee, Brandi Stewart, Courtney JohnsonAuthorsMission Statement:The purpose of this wiki page is to assist in the education of probability and its components to high school students.Relevant Common Core State Standards:What is probability?

In common usage, the word

"probability"is used to mean the chance that a particular event, or set of events, will occur on a scale from 0 to 1.1Basic Probability Terms:

Sample space:the sample space for a given set of events is the set of all possible values the events may assume.Union:("or") the union of two sets A and B is the set containing members of A or B, or both A and B.Intersection:("and") The intersection of two sets A and B is the set of elements common to A and B3Complement:the complement of a set A is all outcomes not in the outcomes of set A. The complement of an event A is denoted Ac and is read "A complement."4 It can be calculated as follows:P(Ac) = 1 - P(A) For example, a card is chosen at random from a deck of cards. What is the probability that a card chosen is not a face card?

Disjoint:two sets are said to be disjoint if their intersection is the empty set, or the two sets have no elements in commonLet's Visualize This!

Independence

S-CP.2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

(See

Multiplication Rule for Independent Eventsbelow)Example: An executive on a business trip must rent a car in each of two different cities. Let A denote the event that the executive is offered a free upgrade in the first city and B represents the analogous event for the second ciry. Suppose P(A)=0.2, P(B)=0.3, P(Ac)=0.8, and P(Bc)=0.7. What is the probability that the executive is offered a free upgrade in at least one of the two cities? *

P(at least one) = 1 - P(Ac and Bc)

=1-[P(Ac)*P(Bc)]

=1-(.8)(.7)

=.44

S-CP.3.** Understand the conditional probability of

Agiven B as P(A and B)/P(B), and interpret independence ofAandBas saying that the conditional probability ofAgivenBis the same as the probability ofA, and the conditional probability ofBgivenAis the same as the probability ofB.For any two events A and B with P(B) > 0, the conditional probability of A given hat B had occurred is defined by:

Independence is when the outcomes of two events A and B do not affect each other. Therefore, independent events do not have conditional probability because the events are not dependent on each other.

Further application....Bayes' Rule!

By definition, Bayes' Rule states:

P(Hk|A) = P(A|Hk) P(Hk) / ∑i P(A|Hi) P(Hi) where the sum is i = 1...n.

Example: A doctor is called to see a sick child. The doctor has prior information that 90% of sick children have the flu, while 10% are sick with measles. F = event that child is sick with the fluM = even that child is sick with measlesA well-known symptom of measles is a rash (denoted by R), P(R|M) = 0.95. However, occasionally children with the flu also develop a rash so that P(R|F) = 0.08. Upon examining the child, the doctor finds a rash. What is the probability that the child has measles?"5Applying the formula, P(M|R) = P(R|M) P(M) / [ (P(R|M) P(M)) + ((P(R|F) P(F))]

=(.95)(.1) / [(.95)(.1) + (.9)(.08)]

=0.5689...= 57%

Rules of Probability

S-CP.7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the uniform probability model.

S-CP.8. Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

Addition Rule:If A and B are separate set of events the then the probability of either event occurring is P(A or B) = P(A) + P(B) - P(A and B)Multiplication Rule for Independent Events:If A and B are independent events then, P(A and B) = P(A) * P(B)8Multiplication Rule for Dependent Events:If A and B are dependent events, then P(A and B) = P(A) * P(B|A)9Permutations and Combinations

S-CP.9. Use permutations and combinations to compute probabilities of compound events and solve problems.

Combinations:Anunorderedsubset of k objects, taken from a set of n distinct objects is a combination of size k***Permutations:anyorderedsequence ofkobjects taken from a set ofndistinct objects***Expected Value

S-MD.1. Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

S-MD.2. Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

S-MD.3. Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

In order to understand the calculation of expected values, it's necessary to understand the concepts of random variables and the probability distribution function.

Random variable:any rule that associates a number with an outcome in the sample space***Probability distribution function:describes probabilities of random variables occurringS-MD.5. Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

Expected value:the weighted average of possible values of a random, where the weights are the probabilities10Sources

- Common Core State Standards:
- http://www.corestandards.org/the-standards/mathematics/hs-statistics-and-probability/conditional-probability-and-the-rules-of-probability/
- http://www.corestandards.org/the-standards/mathematics/hs-statistics-and-probability/using-probability-to-make-decisions/

ImagesThree asterisks (***) denote material from Professor Laura Kapitula's STA 312 Fall 2011 MWF 10-10:50 lecture notes