Probability Monty Hall, Bayes Law, and why Anya probably isn’t autistic Arup Guha 1/14/2013

Probability Basics Sample Space: All possible options – each one MUST BE equally likely!!! Success Space: The items in the sample space defined as “desired” Probability: the number of successes divided by the sample space Example: Simple die roll  We need to roll a 3 or a 5 to win the game.  Sample Space = 6 (we can roll 1, 2, 3, 4, 5 or 6)  Success Space = 2 (3 or 5)  Probability(winning) = 2/6 = 1/3

Probability Pitfalls Incorrect Sample Space Sum of the rolls of two dice has 11 possibilities, 2 – 12 Problem: Not all of these are equally likely.  Chance of rolling a 2 is 1/36, since we must roll two 1s in a roll  Chance of rolling a 7 is 1/6. No matter what we roll at first, one option on the second die allows us to get a sum of 7. Incorrect Counting Given that one die out of two rolled shows a 6, what is the probability that the sum is 8. Clearly two rolls sum to 8 (2, 6) and (6, 2). But how many show at least one 6? If we say that the 6 can appear in two slots and the other slot can have 6 choices, we arrive at 2x6 = 12. But, 11 is correct: (1,6), (2,6), (3,6), (4, 6), (5,6), (6,6), (6, 5), (6,4), (6,3), (6,2), and (6,1).

Pick one of three doors! Monty then reveals one of the other two doors with a goat. You are given the choice to stay with your original choice or switch! Monty Hall – Let’s Make a Deal Door #1Door #2Door #3

Imagine Playing the game 99 times!!! Initial Choice Correct Stay Strategy This will occur roughly 33 times If we stay in these situations, we win! We win 33 times here! Initial Choice Incorrect Stay Strategy This will occur roughly 66 times If we stay in these situations, we lose. We win in none of these situations. We win 33 out of 99 times. This is 1/3.

Imagine Playing the game 99 times!!! Initial Choice Correct Switch Strategy This will occur roughly 33 times If we switch in these situations, we lose  We win 0 times here. Initial Choice Incorrect Switch Strategy This will occur roughly 66 times If we switch in these situations, we win, since Monty was forced to reveal the other goat!!! We win in 66 of these situations! We win 66 out of 99 times. This is 2/3.

Conditional Probability Definition: P(A | B) = probability of A occurring, given that B has occurred. Formula: Intuitively: Once we know that B has occurred, we limit our sample space to include only situations where B happens. Of these, successes are when both A and B occur. Note that we shouldn’t count situations were A occurs but B doesn’t. If A and B are related, then P(A) ≠ P(A|B). Intuitively, knowing about B ought to change our probability of A occurring, if the two items are related!

Example: COP 3223 Spring 2012 Data Sample Space: 175 students Number of A’s: 37 Number of students who spent less than 2 hours per assignment: 13 Number of students of those who spent less than 2 hours per assignment earning an A: 6 Probability earning A = 37/175 ~ 21% Probability earning A given that you spent less than 2 hours per program: = 6/13 ~ 46% Question: What is the probability of earning an A given that you spent 2 or more hours per program?

Example: New Orleans Saints 2012 Overall Record: 7 – 9 Record in games where Drew Brees threw for 300 or more yards: 4 – 6 Record in games where Drew Brees threw for fewer than 300 yards: 3 – 3 P(W) = 7/16 = 43.75% P(W | Brees >= 300 yards) = 4/10 = 40% P(W | Brees < 300 yards) = 3/6 = 50% Counterintuitive result – why? (Teams are forced to throw the ball more when they fall behind in a game.)

Bayes’ Law of Conditional Probability.

Breast Cancer Problem: Bayes’ Law Note: The numbers used for the problem only apply to the population of all women who get tested and not a specific age group. The solution changes as we change our sample size to only be women of certain ages. Set our variables:  P(A) = Probability of having breast cancer  P(B) = Probability of testing positive on a mammography test  P(B | A) = probability of testing positive, given that you have the disease  P(A | B) = probability of having the disease, given that you tested positive for it.

Breast Cancer Problem: Tree Diagram

Breast Cancer Problem - Solution Now, we can see why the AMA doesn’t recommend mammography for women under the age of 40. Many false positives impose a cost on the system and may cause harm and unnecessary worry for many people. For more detail on the AMA position, go to:

Nate Silver and Practical Application of Bayes Use new information (given) to update estimates of existing probabilities In reality, no estimate should be fixed. Rather all estimates ought to be continually updated to reflect new data. Nate has applied these principles to the following fields:  Prediction of Baseball Player Performance  Online Poker  Predicting the outcome of the 2008 and 2012 election

One More Dice Problem Given that the sum of three standard six-sided dice rolled is 6, what is the probability that at least one of the dice shows a 2. Sample Space: (1, 1, 4), (1, 4, 1), (4, 1, 1), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1), and (2, 2, 2). (There are 10 ordered triplets here.) Successes: (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1), and (2, 2, 2). (There are 7 of these.) Probability = 7/10

Last, but not least, Anya! Where is she looking?Child Development Normal Children  Look parents in the eye, usually by one month of age. Autistic Children  Often times, don’t look parents in the eye by one month of age.

Anya and Bayes’ Law My wife, a pediatrician, was worried since at 25 days or so, Anya wasn’t looking us in the eye. Relevant Information  Rate of Autism is believed to be 1.1% (1 in 90)  We need to know what percentage of normal kids don’t look their parents in the eye by one month.  We need to know what percentage of kids with Autism do look their parent in the eye by one month. Rough Bayes Law estimate  If even 15% percent of normal kids are below the curve, then, the probability that a kid has autism given that they haven’t looked their parent in the eye by day 25 is rather low! (Less than10%)