Probability The Study of Chance! Part 3

In this powerpoint we will continue our study of probability by looking at: In this powerpoint we will continue our study of probability by looking at: Conditional ProbabilityConditional Probability IndependenceIndependence Reversing the ConditionReversing the Condition

Conditional Probabilities What about when we know something up front about the individual that we have chosen? What about when we know something up front about the individual that we have chosen? How does this change our calculations? How does this change our calculations? Let’s look a the employment/credit card data that we used in Part 2. Let’s look a the employment/credit card data that we used in Part 2.

Conditional Probabilities Total No Yes TotalNoYesJOB Credit Card What is the probability that a respondent has a credit card given that they are employed? We write this type of question like this: P(C E) *the line between is read “given” and indicates that we already know the outcome of the event that comes after the line. Since we already know that the respondent has a job, this means that our population of interest is only the 47 respondents that have jobs. 18 of those have credit cards. So P(C E) = 18/47

Conditional Probability Find the probability that a respondent does not have a job given that they have a credit card. Find the probability that a respondent does not have a job given that they have a credit card. P(E c C) = P(E c C) = Since we already know they have a credit card, our population of interest has become they 46 individuals who have a credit card.Since we already know they have a credit card, our population of interest has become they 46 individuals who have a credit card. Our “successes” are the 28 of those people who do not have a job.Our “successes” are the 28 of those people who do not have a job. P(E c C) = 28/46 P(E c C) = 28/46 Total No Yes TotalNoYesJOB Credit Card

Disjoint vs Independent Two very important concepts in probability are disjoint events and independent events. The ability to tell the difference between these are very important when calculating probabilities. Two very important concepts in probability are disjoint events and independent events. The ability to tell the difference between these are very important when calculating probabilities.

Disjoint Events (Mutually Exclusive) We’ll start with disjoint events. We’ll start with disjoint events. When events are disjoint this means they have no outcomes in common.When events are disjoint this means they have no outcomes in common. Mutually Exclusive is another term used to describe disjoint events.Mutually Exclusive is another term used to describe disjoint events. Examples: Examples: The relationships between disjoint and joint events may be best shown using Venn diagrams.The relationships between disjoint and joint events may be best shown using Venn diagrams. Let the event A: Democratic voterLet the event A: Democratic voter Let the event B: Republican voterLet the event B: Republican voter Democratic Voter A B Republican Voter Since a voter can not be registered as both a Democratic Voter and a Republican Voter, there is no overlap of the events. This can either be shown with non-overlapping Venn diagrams or with the overlap being empty.

Independence Events are independent when knowing the outcome of one event does NOT change the probability of the other. Events are independent when knowing the outcome of one event does NOT change the probability of the other. For Example: For Example: Consider the two events: flip a coin, then roll a dieConsider the two events: flip a coin, then roll a die We wish to know the probability of the eventWe wish to know the probability of the event Head and a number less than 2 Head and a number less than 2 The probability of getting heads on a fair coin is ½The probability of getting heads on a fair coin is ½ The probability of getting a number less than 2 on a fair die is 2/6 or 1/3The probability of getting a number less than 2 on a fair die is 2/6 or 1/3 Knowing the outcome of the coin toss does NOT change the probability of getting a number less than 2 on the die.Knowing the outcome of the coin toss does NOT change the probability of getting a number less than 2 on the die. These two events are independent, since knowing the outcome of the coin toss didn’t change the probability of the dice roll.These two events are independent, since knowing the outcome of the coin toss didn’t change the probability of the dice roll.

Independence The formal mathematical definition tells us that two events are independent if: The formal mathematical definition tells us that two events are independent if: P(A B) = P(A) P(A B) = P(A) In other words, this “insider” information (knowing that “B” has happened doesn’t gives us NO new information about “A” happening! In other words, this “insider” information (knowing that “B” has happened doesn’t gives us NO new information about “A” happening! Let’s look at an example. Let’s look at an example.

Tests for Drunk Driving Police report that 78% of drivers are given a breath test, 36% a blood test, and 22% both tests Police report that 78% of drivers are given a breath test, 36% a blood test, and 22% both tests We wish to know if giving the two test are independent? We wish to know if giving the two test are independent? Let’s construct a contingency table for this situation Let’s construct a contingency table for this situation Breath Test YesNoTotal Blood Test Yes No Total

Independent???? Breath Test YesNoTotal Blood Test Yes No Total If the two tests are independent then the following should be true: P(blood test breath test ) = P(blood test) Let’s find out if they are.

Breath Test YesNoTotal Blood Test Yes No Total P(blood test breath test ) = P(blood test) Remember from our study of conditional probabilities that the P(blood test breath test) =.22/.78=.28 P(blood test) =.36 Since.28 ≠.36, these events are NOT independent. Knowing the outcome of one gives changes the probability of the other

Disjoint vs Independent Can disjoint events ever be independent? Can disjoint events ever be independent? Remember that for events to be independent, knowing the outcome of one does NOT change the outcome of the other.Remember that for events to be independent, knowing the outcome of one does NOT change the outcome of the other. Also remember that disjoint events can NOT happen at the same time, which means knowing that one happened makes the probability of the other happening zero (regardless of its original probability)Also remember that disjoint events can NOT happen at the same time, which means knowing that one happened makes the probability of the other happening zero (regardless of its original probability) In short: Disjoint events can NEVER be independent!!!!! In short: Disjoint events can NEVER be independent!!!!!

Reversing the condition Sometimes the information we know and the information we are looking for doesn’t come in such a way as to be a straight forward question. Sometimes the information we know and the information we are looking for doesn’t come in such a way as to be a straight forward question. Let’s find out how we can use tree diagrams to answer these kinds of questions. Let’s find out how we can use tree diagrams to answer these kinds of questions.

Binge drinking and Accidents Our question of interest is this: Our question of interest is this: What is the probability that if a college student has been involved in an alcohol-related accident, s/he is a binge drinker?What is the probability that if a college student has been involved in an alcohol-related accident, s/he is a binge drinker? Here is what we know: Here is what we know: A study on binge drinking on Campus found that 44% of college students engage in binge drinking (5+ for men, and 4+ for women), 37% drink moderately, and 19% abstain entirely.A study on binge drinking on Campus found that 44% of college students engage in binge drinking (5+ for men, and 4+ for women), 37% drink moderately, and 19% abstain entirely. Another study found that among binge drinkers aged 21-34, 17% have been involved in an alcohol-related accident, while among non-bingers of the same age, only 9% have been involved in such accidents.Another study found that among binge drinkers aged 21-34, 17% have been involved in an alcohol-related accident, while among non-bingers of the same age, only 9% have been involved in such accidents. Let’s draw a tree diagram to help answer our question Let’s draw a tree diagram to help answer our question

Binge Drinkers.44 Moderate.37 Abstain.19 Accident.17 Accident.09 Accident.00 None.83 None.91 None 1.0 Binge and Acc.44(.17)=.0748 Binge and None.44(.83)=.3652 Mod and Acc.37(.09)=.0333 Mod and None.37(.91)=.3367 Abstain and Acc.19(.00) = 0 Abstain and None.19(1.0) =.19 Remember that our question can be written: P(binge accident) What we know is the second event, (accident). This essentially changes our sample space (our denominater) We are only concerned with the branches where accidents occurred. Our numerator is the branch among these in which the driver is a binge drinker. P(Binge Acc) =.0748 / ( ) =.0748/.1081 =.6920

What you learned Disjoint Events Disjoint Events Independent Events Independent Events Disjoint vs Independent Disjoint vs Independent Reversing the Condition Reversing the Condition

Additional Resources The Practice of Statistics—YMM The Practice of Statistics—YMM Pg Pg The Practice of Statistics—YMS The Practice of Statistics—YMS Pg Pg Against All Odds—Video #15 Against All Odds—Video # htmlhttp:// 65.htmlhttp:// 65.htmlhttp:// 65.html