Tree Diagrams•Tree diagrams help us think through conditional probabilities by showing sequences of events as paths that look like branches of a tree •We often make tree diagrams when reversing the conditioning •Suppose we want to know Prob(A | B), but we know only Prob(A), Prob(B) and Prob(B | A) •We also know Prob(A and B), since P(A and B) = Prob(A) x Prob(B | A) •From this information, we can find Prob(A | B) When we reverse the probability from the conditional probability that we are originally give, we use Bayes TheoremExample - False Positive Rates•Assume there is a screening test for a certain cancer that is 95 percent accurate if someone has the cancer. Also assume that if someone doesn't have the cancer, the test is positive just 1 percent of the time. Assume further that 0.5 percent actually have this type of cancer. What is the probability that someone who tested positive for this cancer does not actually have the cancer, i.e. what is the false positive rate? P(+/C) = .95P(+/NC) = .01P(C) = .005P(NC/+) = ?=P(NC and +)/P(+)conditionalExample - False Positive Ratessums to 1multiply acrossExample - False Positive Rates•Using Bayes Rule:•Approximately 68% of people who test positive for cancer do not actually have cancer!Example - False Positive Rates•What percent of people who test positive for cancer actually have cancer?P(C/+) = .00475/ .00475+.00995 = .32Example - HIV Test•HIV prevalence is .006 in the US population, so .994 do not have HIV. There is a HIV test that if you have the disease 99% of the time the test says positive (1% false negative). If you don't have the disease 98% of the time the test says negative (2% false positive). What is the probability that someone actually has HIV if the test says positive? .994.98.02.006.99.01.97412.01988.00594.00006No HIVYes HIVP(HIV/+) = P(HIV and +) / P(+)(.00594)/ (.00594 + .01988) = .23Chapter 6: Modeling Random Events - The Normal and Binomial ModelsProbability Model and Distribution•A probability model is a description of how a statistician thinks data are produced •Uniform •Linear •Normal •Other •A probability distribution or probability distribution function (pdf) is a table or graph that gives all the outcomes of a random experiment and their probabilitiesDiscrete vs Continuous•A random variable is called discrete if the outcomes are values that can be listed or counted •Number of classes taken •The roll of a die •A random variable is called continuous if the outcomes cannot be listed because they occur over a range •Time to finish the exam •Exact weightDiscrete vs ContinuousClassify the following as discrete or continuous •Length of your left thumb •Number of children in a family •Number of devices in the house that connect to the Internet •Sodium concentration in the bloodstream cDDCDiscrete Probability Distributions •The most common way to display a pdf for discrete data is with a table •The probability distribution table always has two columns (or rows) •The first, x, displays all the possible outcomes •The second, P(x), displays the probabilities for these outcomesExamples of Probability DistributionsImportant: The sum of all the probabilities must equal 1Example - Playing Dice•Roll a fair six-sided die. You will win $4 if you roll a 5 or a 6. You will lose $5 if you roll a 1. You will lose $1 if you roll a 2. Any other outcome, you will win or lose $0. What is the probability distribution table for the amount you will win? 1 = -$52 = -$23 = $04 = $05 = +$46 = +$44 = p 2/60 = p 2/6-1 = p 1/6-5 = p 1/6The Binomial Model•The binomial probability distribution is a discrete probability distribution function •Useful in many situations where you have numerical variables that are counts or whole numbers•Surprise! Classic application of the binomial model is counting heads when flipping a coinThe Binomial Model•The binomial model provides probabilities for random experiments in which you are counting the number of successes that occur. Four characteristics must be present: 1) Fixed number of trials: n2) The only two outcomes are success and failure3) The probability of success, p, is the same at each trial 4) The trials are independentBinomial or Not?•40 randomly selected college students were asked if they selected their major in order to get a good job. •35 randomly selected Americans were asked what country their mothers were born. •To estimate the probability that students will pass an exam, the professor records a study group's success on the exam. yesnono, not independentComputing Binomial ProbabilitiesA Stats 10 test has 4 multiple choice questions with four choices with one correct answer each. If we just randomly guess on each of the 4 questions, what is the probability that you get exactly 3 questions correct? •There are 4 different outcomes in which you could get 3 of 4 questions correct: Correct, Correct, Correct, Wrong Correct, Correct, Wrong, Correct Correct, Wrong, Correct, Correct Wrong, Correct, Correct, CorrectComputing Binomial ProbabilitiesA Stats 10 test has 4 multiple choice questions with four choices with one correct answer each. If we just randomly guess on each of the 4 questions, what is the probability that you get exactly 1 question correct? Correct, Correct, Correct, Wrong = 0.25 x 0.25 x 0.25 x 0.75= 0.01172 •The four outcomes all have the same probability so the probability that you get exactly 3 correct is 4 x 0.01172 = 0.04668Computing Binomial ProbabilitiesA Stats 10 test has 4 multiple choice questions with four choices with one correct answer each. If we just randomly guess on each of the 4 questions, what is the probability that you get exactly 1 question correct? a) 0.04668 b) 0.42188 c) 0.10547 d) 0.25 4x(.25)(.75)^3Binomial Distribution Function•The formula that finds the probabilities for the binomial distribution for probability of success p, fixed number of trials n, and k successes is as follows: k = number of sucessesBinomial Coefficient•The n over the k inside the parentheses can be read as “n choose k” •Instead of writing all different combinations of outcomes and counting them all one-by-one this provides us the number of all those combinations.Factorials•! - indicates a factorial•n! = n x (n-1) x (n-2) x (n-3) x .... x 1 5! = 5 x 4 x 3 x 2 x 1 = 1206! = 6 x 5 x 4 x 3 x 2 x 1 = 720Binomial Coefficient ExamplesBinomial Coefficient HintsComputing Binomial Probabilities•A