1. Chapter 5: Probability in Our Daily Lives Section 5.4: Applying the Probability Rules

2. Is a “Coincidence” Truly an Unusual Event? The law of large numbers states that if something has a very large number of opportunities to happen, occasionally it will happen, even if it seems highly unusual What is the probability that at least two students in a group of 25 students have the same birthday? P(at least one match) = 1 – P(no matches) P(no matches) = P(students 1 and 2 and 3 …and 25 have different birthdays) P(no matches) = (364/365) x (363/365) x (362/365)….(341/365) Student 2 doesn’t match given 1 Student 3 doesn’t match given 1,2 Student 4 doesn’t match given 1,2,3 Student 25 doesn’t match given 1,2,3,…24 P(no matches) =.43 P(at least one match) = 1 – P(no matches) = 1 -.43 =.57 = 57%

3. Probability Model We’ve dealt with finding probabilities in many idealized situations In practice, it’s difficult to tell when outcomes are equally likely or events are independent In most cases, we must specify a probability model that approximates reality A probability model specifies: the possible outcomes for a sample space and provides assumptions on which the probability calculations for events composed of these outcomes are based Probability models merely approximate reality

4. Out of the first 113 space shuttle missions there were two failures What is the probability of at least one failure in a total of 100 missions? P(at least 1 failure)=1-P(All successes) One study found that the probability of success of a space mission is P(S) =.971 =.053 P(S1 and S2 and S3…and S100) = P(S1) x P(S2) x P(S3)…x P(S100) P(S1 and S2 and S3…and S100) =.971 x.971 x.971…x.971 P(at least 1 failure) = 1 -.053 =.947 =94.7% This answer relies on the assumptions of Same probability of success on each flight Independence These assumptions are suspect since other variables (temperature at launch, crew experience, age of craft, etc.) could affect the probability

5. Probabilities and Diagnostic Testing Sensitivity = P(POS|S)-Is the probability that the test is positive given that the state is truly present (Correct positive test) Specificity = P(NEG|S C )-Is the probability that the test is negative given that the state is truly absent (Correct negative test) Prevalence = P(S)-The probability that the state is present

6. Triple Blood Test: Triple Blood Test for Down syndrome found the results shown in the table below. a)Estimate the Prevalence (Down = Yes) b) Find the estimated sensitivity= P(POS|S) c) Find the estimated specificity= P(NEG|S C ) Blood Test results DownPOSNEGTotal Yes48654 No130739215228 Total135539275282 54/5282 =.0102 =1.02%

7. Simulatio n Carrying out a Simulation: Identify the random phenomenon to be simulated Describe how to simulate observations Carry out the simulation many times (at least 1000 times) Summarize results and state the conclusion Some probabilities are very difficult to find with ordinary reasoning. In such cases, we can approximate an answer by simulation. 1) Simulation Free throws for an NBA player who makes them 88% of the time. 2) Use random number generator #1-88 simulate a made free throw #89-100 simulate a miss 3) Generate 1000 random numbers from 1 to 100 and tally the “makes” and “misses” 4) Summarize your results