Probability, Bayes Theorem and the Monty Hall Problem

Probability DistributionsA random variable is a variable whose value is uncertain.For example, the height of a randomly selected person in this class is a random variable I wont know its value until the person is selected.Note that we are not completely uncertain about most random variables. For example, we know that height will probably be in the 5-6 range. In addition, 56 is more likely than 50 or 60 (for women).The function that describes the probability of each possible value of the random variable is called a probability distribution.

Probability DistributionsProbability distributions are closely related to frequency distributions.

Probability DistributionsDividing each frequency by the total number of scores and multiplying by 100 yields a percentage distribution.

Probability DistributionsDividing each frequency by the total number of scores yields a probability distribution.

Probability DistributionsFor a discrete distribution, the probabilities over all possible values of the random variable must sum to 1.

Probability DistributionsFor a discrete distribution, we can talk about the probability of a particular score occurring, e.g., p(Province = Ontario) = 0.36.We can also talk about the probability of any one of a subset of scores occurring, e.g., p(Province = Ontario or Quebec) = 0.50.In general, we refer to these occurrences as events.

Probability DistributionsFor a continuous distribution, the probabilities over all possible values of the random variable must integrate to 1 (i.e., the area under the curve must be 1).Note that the height of a continuous distribution can exceed 1!

Continuous DistributionsFor continuous distributions, it does not make sense to talk about the probability of an exact score.e.g., what is the probability that your height is exactly 65.485948467 inches?Normal Approximation to probability distribution for height of Canadian females(parameters from General Social Survey, 1991)

Continuous DistributionsIt does make sense to talk about the probability of observing a score that falls within a certain rangee.g., what is the probability that you are between 53 and 57?e.g., what is the probability that you are less than 510?Normal Approximation to probability distribution for height of Canadian females(parameters from General Social Survey, 1991)

Probability of Combined Events

Probability of Combined Events

Exhaustive EventsTwo or more events are said to be exhaustive if at least one of them must occur.For example, if A is the event that the respondent sleeps less than 6 hours per night and B is the event that the respondent sleeps at least 6 hours per night, then A and B are exhaustive.(Although A is probably the more exhausted!!)

Independence

An Example: The Monty Hall Problem

Problem HistoryWhen problem first appeared in Parade, approximately 10,000 readers, including 1,000 PhDs, wrote claiming the solution was wrong.In a study of 228 subjects, only 13% chose to switch.

IntuitionBefore Monty opens any doors, there is a 1/3 probability that the car lies behind the door you selected (Door 1), and a 2/3 probability it lies behind one of the other two doors.Thus with 2/3 probability, Monty will be forced to open a specific door (e.g., the car lies behind Door 2, so Monty must open Door 3).This concentrates all of the 2/3 probability in the remaining door (e.g., Door 2).

AnalysisSwitching loses with probability 1/6Switching wins with probability 2/3Switching loses with probability 1/3Switching wins with probability 1/3Switching wins with probability 1/3Switching loses with probability 1/6Host must open Door 2Host must open Door 3Host opens either Door 2 or 3Player initially picks Door 1Car hidden behind Door 3Car hidden behind Door 2Car hidden behind Door 1

NotesIt is important thatMonty must open a door that reveals a goatMonty cannot open the door you selectedThese rules mean that your choice may constrain what Monty does.If you initially selected a door concealing a goat, then there is only one door Monty can open.One can rigorously account for the Monty Hall problem using a Bayesian analysis

End of Lecture 2Sept 17, 2008

Conditional ProbabilityTo understand Bayesian inference, we first need to understand the concept of conditional probability.What is the probability I will roll a 12 with a pair of (fair) dice?What if I first roll one die and get a 6? What now is the probability that when I roll the second die they will sum to 12?

Conditional ProbabilityThe conditional probability of A given B is the joint probability of A and B, divided by the marginal probability of B.

Thus if A and B are statistically independent,

However, if A and B are statistically dependent, then

Bayes TheoremBayes Theorem is simply a consequence of the definition of conditional probabilities:

Bayes TheoremBayes theorem is most commonly used to estimate the state of a hidden, causal variable H based on the measured state of an observable variable D:

Bayesian InferenceWhereas the posterior p(H|D) is often difficult to estimate directly, reasonable models of the likelihood p(D|H) can often be formed. This is typically because H is causal on D.Thus Bayes theorem provides a means for estimating the posterior probability of the causal variable H based on observations D.

MarginalizingTo calculate the evidence p(D) in Bayes equation, we typically have to marginalize over all possible states of the causal variable H.

The Full MontyLets get back to The Monty Hall Problem.Lets assume you initially select Door 1.Suppose that Monty then opens Door 2 to reveal a goat. We want to calculate the posterior probability that a car lies behind Door 1 after Monty has provided these new data.

The Full Monty

The Full Monty

But were not on Lets Make a Deal!Why is the Monty Hall Problem Interesting?It reveals limitations in human cognitive processing of uncertaintyIt provides a good illustration of many concepts of probabilityIt get us to think more carefully about how we deal with and express uncertainty as scientists.What else is Bayes theorem good for?

Clinical ExampleChristiansen et al (2000) studied the mammogram results of 2,227 women at health centers of Harvard Pilgrim Health Care, a large HMO in the Boston metropolitan area. The women received a total of 9,747 mammograms over 10 years. Their ages ranged from 40 to 80. Ninety-three different radiologists read the mammograms, and overall they diagnosed 634 mammograms as suspicious that turned out to be false positives.This is a false positive rate of 6.5%.The false negative rate has been estimated at 10%.

Clinical ExampleThere are about 58,500,000 women between the ages of 40 and 80 in the USThe incidence of breast cancer in the US is about 184,200 per year, i.e., roughly 1 in 318.

Clinical Example