Probability
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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.
union.png
    • Pronounced "A union B"
    • Can be calclated as such:
    • unioncalc.png
  • Intersection: ("and") The intersection of two sets A and B is the set of elements common to A and B3

intersection.png
    • 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...
    • Intersection:
    • external image VennDiagram_900.gif



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:
conditional.png
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.
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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

Applying 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)
    • 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:
    • combination.png
    • Example: If you have a deck of cards without the jokers, how many 5-card hands can you get?
    • combination52.png = 2,598,960


  • Permutations: any ordered sequence of k objects taken from a set of n distinct objects***
    • Notation:
    • Permutation.PNG
    • 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? ***
    • example.PNG

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.
      • S = { (D,D,D), (D,D,F), (D,F,F), (F,F,F), (D,F,D), (F,F,D), (F,D,F), (F,D,D) }
      • X = 3, 2, 1, 0, 2, 1, 1, 2

  • 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:
    • pdf.pngpdfchart.png
    • 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
  • expected_value.png
  • 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.***
  • pdfev.png
  • (a) Compute E(X)
  • pdfev3.png
  • (b) Suppose an individual with X violations incurs a surcharge of $100X2. Calculate the expected amount of the surcharge.
  • pdfev2.png
  • pdfev4.png




Sources

Images

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