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Modeling Variation with Probability. Upcoming Assignments. Ch5 Reading Quiz – due 10/18 Ch5 Activity Ch5 HW Ch5 Book HW Exam 2. Learning Objectives. Two Types of Probabilities Finding Theoretical Probabilities Associations in Categorical Variables
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Upcoming Assignments • Ch5 Reading Quiz – due 10/18 • Ch5 Activity • Ch5 HW • Ch5 Book HW • Exam 2
Learning Objectives • Two Types of Probabilities • Finding Theoretical Probabilities • Associations in Categorical Variables • Empirical Probabilities – Law of Large Numbers
Randomness • Randomness indicates that no predictable pattern occurs in an events. A sequence of random events has no order and is lack of intelligible pattern. • Having flipped a coin 5 times in a row and gotten heads on each flip doesn’t impact the probability of the coin landing on heads at the next flip.
Two Types of Probabilities • Probability is used to measure how often random events occur. In this chapter, we will examine two kinds of probabilities: Empirical Probabilities and Theoretical Probabilities. • Theoretical Probabilities are long-run relative frequencies based on theory. • ex: a coin has 50% probability of landing on heads • Empirical Probabilities are short term relative frequencies based on experiments. • ex: Out of 20 times coin flips, 14 landed on heads. The probability of the coin landing on heads is 14/20=70%.
Exercise: Empirical or Theoretical Identify each as an example of an empirical or a theoretical probability. • A monopoly player claims that the probability of getting a 4 when rolling a 6-sided die is one-sixth since the die is equally likely to land on any of the 6 sides. • Out of the 100 times I shopped in Amazon, 1 time a wrong item was sent. The probability of receiving a wrong item when shopping in Amazon is 1%.
Some Facts and Terminologies • Probabilities are ALWAYS within 0 and 1. • P(A) denotes the probability of the event A. • ex: P(heads) denotes the probability of the coin landing on heads. • A “NOT event ” is called a complement of the event , denoted as . The probability of a not event will occur equals -the probability of the event occurs. • ex: Denote the event as the coin landing on heads. What is the complement of the event A in words? • Equally likely outcomes mean each outcome has the same probability of occurring. • ex: Heads and tails are equally likely outcomes for a two-sided coin. • P(A) =
Exercise: • Ex1: A multiple choice question has 5 possible answers, only two of which are correct. What is the probability of guessing incorrectly on the question? • Ex2: Reach into a bowl that contains 5 red dice, 3 green dice, and 2 white dice. What is the probability of picking a green die?
Combining Events • The words AND and OR can be used to combine events into new events. • Venn Diagram • The event of what you are good at and what you love • The event of what you are good at or what you love • Which event is a larger set?
Exercise: AND/OR Events Picking a number from 1-10. a. What numbers are both even and less than 5? b. Randomly select a number from 1-10, what is the probability of choosing an even number that is also less than 5? c. What numbers are either even or less than 5? d. Randomly select a number from 1-10, what is the probability of choosing an even number or a number that is less than 5?
AND/OR Events Notations • Denote the set of even numbers as event A and the set of numbers less than 5 as event B. • denotes the probability of the event A or B. • denotes the probability of the event A and B. • Relation between the AND and OR events: . • ex: Using the number exercise to examine the relation.
Exercise: Highest Educational Level and Current Marital Status Randomly Select a person. a. What is the probability that person is single? b. What is the probability that the person has a college education and is single? c. What is the probability that the person is single or is married? d. What is the probability that the person is married or has a high school degree?
Mutually Exclusive Events • Referring to parts c and d, what do you observe? • Two events are called mutually exclusive if both events won’t occur at once. • A person can’t be both single and married at once. • A person can’t have a high school and a college degree as the highest educational level. • Name two mutually exclusive events in the previous example. • if events A and B are mutually exclusive. • whenevents A and B are mutually exclusive. (part c.)
Exercise: Assume the only grades possible in the Intro Stats course are A, B, C, or lower than C. The probability that a randomly selected student will get an A in the course is 0.32, the probability that a student will get a B in the course is 0.37, and the probability that a student will get a C in the course is 0.13. • What is the probability that a student will get an A OR a B? • What is the probability that a student will get an A OR a B OR a C? • What is the probability that a student will get a grade lower than a C?
Associations • In the previous chapter, we summarize two categorical variables by a two-way table. Often we are interested in investigating the association between the two variables. • Recall the two-way table with the highest educational degree and the marital status for a collection of 665 people, if there is an association between the two variables, we would expect the probability that a randomly selected college-educated person is married to be different from the probability that a person with less than a college education is married.
Conditional Probabilities • Conditional probability is a measure of the probability of an event given that another event has occurred. • the probability that a randomly selected college-educated person is married. • the probability that a person with less than a college education is married. • denotes the probability of A occurring given that B has occurred. • P ( Married | college-educated ) • P ( Married | less than college education )
Calculating Conditional Probabilities • Trick to calculate such probabilities is to focus on the given group of objects (condition) and imagine taking a random sample from the group. ex: the probability that a randomly selected college-educated person is married: • Focus on the group of college-educated people, calculate the probability of selecting one that is married. ex: the probability that a person with less than a college education is married: • Focus on the group of _____________ people, calculate the probability of selecting one that is married.
Exercise: Highest Educational Level and Current Marital Status Randomly Select a person. a. What is the probability that college-educated person is married? b. What is the probability that a person with less than a college education is married?
Exercise: Highest Educational Level and Current Marital Status Randomly Select a person. c. What is the probability that a married person has a high school degree? d. What is the probability that a divorced person has a less than high school?
AND Events and Conditional Probabilities • The formula for calculating conditional probabilities is . • Multiplying on the both sides, you will get. • You are not required to memorize these two forms but it’s a good practice to redo the previous exercise using the formula.
Independent Events • In the previous examples, we observe that 59% of people with less than a college education is married while 69% of people with at least a college education is married. In other words, people with a college education has a higher marriage rate. • Another way of putting this is to say the marital status and the education level are associated, i.e. dependent.
Independent Events • Had we observed 59% of people with less than a college education is married and 59% of people with at least a college education is married, it tells us that regardless of the education level, the percent of people being married is 59%. • In other words, the marital status and the education level are independent when P ( Married | less than college education ) = P ( Married | college education ) =P ( Married ) • Events A and B are independent if P ( A | B ) = P ( A )
Exercise: Draw a card form a standard 52 cards. Are events drawing a diamond card and drawing a queen independent ? Draw a card form a standard 52 cards. Are events drawing a even number card and drawing a 5 independent ?
Sequence of Independent Events • Events that take place in a certain order are called sequence of events. • Planning to have 2 kids, having the 1st as a girl and the 2nd as a boy • Flipping a coin 5 times, the coin landing as HTHTH. • Multiplication Rule: When events A and B are independent, . ex: Flipping a coin 5 times, the probability of the coin landing as HTHTH is ex: Flipping a coin 5 times, the probability of the coin landing as HHHHH is
Exercise: The percentage of Americans over the age of 18 that drink coffee daily is 54%. Randomly select two adult Americans. • Find the probability that both drink coffee daily. • Find the probability that neither drinks coffee daily. • Find the probability that at least one of them drinks coffee daily.
Exercise: According to a 2012 survey, 44% of U.S. public school teachers reported that they were satisfied with their jobs. Suppose we select three teachers randomly with replacement. • Find the probability that all three are satisfied with their careers. • Find the probability that none of them is satisfied with their careers. • Find the probability that at least one of them is satisfied with their careers.
