Probability, Bayes’ Theorem and the Monty Hall Problem

Probability Distributions A 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 won’t 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, 5’6” is more likely than 5’0” or 6’0” (for women). The function that describes the probability of each possible value of the random variable is called a probability distribution. PSYC 6130, PROF. J. ELDER

Probability Distributions Probability distributions are closely related to frequency distributions. PSYC 6130, PROF. J. ELDER

Probability Distributions Dividing each frequency by the total number of scores and multiplying by 100 yields a percentage distribution. PSYC 6130, PROF. J. ELDER

Probability Distributions Dividing each frequency by the total number of scores yields a probability distribution. PSYC 6130, PROF. J. ELDER

Probability Distributions For a discrete distribution, the probabilities over all possible values of the random variable must sum to 1. PSYC 6130, PROF. J. ELDER

Probability Distributions For 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. PSYC 6130, PROF. J. ELDER

Probability Distributions For 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! S h a d e r = . 6 8 3 9 5 4 7 PSYC 6130, PROF. J. ELDER

Continuous Distributions For 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) 55 60 65 70 75 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Height (in) Probability p ? PSYC 6130, PROF. J. ELDER

Continuous Distributions It does make sense to talk about the probability of observing a score that falls within a certain range e.g., what is the probability that you are between 5’3” and 5’7”? e.g., what is the probability that you are less than 5’10”? Valid events Normal Approximation to probability distribution for height of Canadian females (parameters from General Social Survey, 1991) 55 60 65 70 75 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Height (in) Probability p PSYC 6130, PROF. J. ELDER

Probability of Combined Events PSYC 6130, PROF. J. ELDER

Probability of Combined Events PSYC 6130, PROF. J. ELDER

Exhaustive Events Two 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!!) PSYC 6130, PROF. J. ELDER

Independence PSYC 6130, PROF. J. ELDER

An Example: The Monty Hall Problem PSYC 6130, PROF. J. ELDER

Problem History When 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. PSYC 6130, PROF. J. ELDER

Intuition Before 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). PSYC 6130, PROF. J. ELDER

PSYC 6130, PROF. J. ELDER

Analysis Car hidden behind Door 1 Car hidden behind Door 2 Player initially picks Door 1 Host opens either Door 2 or 3 Host must open Door 3 Host must open Door 2 Switching loses with probability 1/6 Switching loses with probability 1/6 Switching wins with probability 1/3 Switching wins with probability 1/3 Switching loses with probability 1/3 Switching wins with probability 2/3 PSYC 6130, PROF. J. ELDER

Notes It is important that Monty must open a door that reveals a goat Monty cannot open the door you selected These 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 PSYC 6130, PROF. J. ELDER

End of Lecture 2 Sept 17, 2008

Conditional Probability To 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? “Probability of C given A” PSYC 6130, PROF. J. ELDER

Conditional Probability The 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 PSYC 6130, PROF. J. ELDER

Bayes’ Theorem Bayes’ Theorem is simply a consequence of the definition of conditional probabilities: Bayes’ Equation PSYC 6130, PROF. J. ELDER

Bayes’ Theorem Bayes’ 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: Likelihood Prior Posterior Evidence PSYC 6130, PROF. J. ELDER

Bayesian Inference Whereas 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. PSYC 6130, PROF. J. ELDER

Marginalizing To calculate the evidence p(D) in Bayes’ equation, we typically have to marginalize over all possible states of the causal variable H. PSYC 6130, PROF. J. ELDER

The Full Monty Let’s get back to The Monty Hall Problem. Let’s 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. PSYC 6130, PROF. J. ELDER

The Full Monty PSYC 6130, PROF. J. ELDER

The Full Monty PSYC 6130, PROF. J. ELDER

But we’re not on Let’s Make a Deal! Why is the Monty Hall Problem Interesting? It reveals limitations in human cognitive processing of uncertainty It provides a good illustration of many concepts of probability It get us to think more carefully about how we deal with and express uncertainty as scientists. What else is Bayes’ theorem good for? PSYC 6130, PROF. J. ELDER

Clinical Example Christiansen 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%. PSYC 6130, PROF. J. ELDER

Clinical Example There are about 58,500,000 women between the ages of 40 and 80 in the US The incidence of breast cancer in the US is about 184,200 per year, i.e., roughly 1 in 318. PSYC 6130, PROF. J. ELDER

Clinical Example PSYC 6130, PROF. J. ELDER