Comment on the “Tree Diagrams” SectionThe reversal of conditional probabilities when using tree diagrams(calculating P (B | A) from P (A | B) and P (A | Bc)) is an ex-ample of Bayes’ formula, named after the 18th century Englishclergyman Rev. Thomas Bayes. His formula is the foundationfor a whole theory of Statistics called Bayesian statistics.In common-English discussions of statistics and probability, con-fusing P (B | A) and P (A | B) is one of the most common mis-takes people make! Bayes’ formula provides a resolution of thatissue.1Probability DistributionsRandom Variable: A numerical measurement of the outcome ofa random phenomenon.Probability Distribution: All possible values and their probabili-ties.Discrete if it’s possible to express the possible values as a list.Not all random variables discrete — main alternative continuous.If X is a discrete random variable (capital letters for the name ofthe variable) taking values x (small letters for actual numericalvalues) then the probability distribution is given by specifying thevalues P (x), where this has the interpretation “the probabilitythat X is equal to x”. For this to be valid, we require that P (x)is between 0 and 1 for each possible x, and that the sum of P (x)over all x is 1.2Example 1: Let X be the total number of spots in one throw ofa die. Then the possible values of x are 1,2,3,4,5,6. In this caseP (1) = P (2) = P (3) = P (4) = P (5) = P (6) =16.Example 2: Let X be the total number of home runs the Red Soxhit in a game. In this case the possible values x are 0,1,2,3,4,5,6,7,8,....— there’s no actual limit to how many home runs they couldhit, but we can still arrange the possible values in a list so it’sa discrete random variable. According to some data collectedduring the 2004 season, the probabilities are:x P (x) x P (x)0 0.23 4 0.031 0.38 5 0.012 0.22 ≥ 6 03 0.13In this case the numbers P (x) are all between 0 and 1 and doadd up to 1, so they are a consistent probability distribution.3Example 3: Let X be the number of games that UNC win in theirnext football season. Since this represents a single outcomerather than some experiment that’s repeated many time, anyprobabilities we might assess for this would have to be subjectiveprobabilities. However the concept of a probability distributionand how we might manipulate the numbers are exactly the sameas in the example just discussed.4We’ll use Example 2 to illustrate how probability distributionsare used. The general idea is to use given values of P (x) tocalculate probabilities of more complicated events.• Question: “What is the probability the Red Sox score at least3 home runs in a single game?”• Answer: 0.13+0.03+0.01=0.17.• Question: “What is the probability the number of home runsis less than two?”.• Answer: 0.23+0.38=0.61.Be careful about the correct usage of phrases such as “at least3” (the number 3 is included) or “less than 2” (the number 2 isexcluded).5We could also ask “What is the average number of home runs theRed Sox hit in a game?” We could give a “relative frequency”calculation by imagining a hypothetical season of exactly 100games in which the Sox score home runs in exactly the proportiongiven by the above probability distribution, i.e. 23 games theyscore none, 38 games they score 1, and so on. Then the totalnumber of runs scored is(0 × 23) + (1 × 38) + (2 × 22) + (3 × 13) + (4 × 3) + (5 × 1)= 38 + 44 + 39 + 12 + 5 = 138,which is an average of 1.38 home runs per game.6Alternatively, we could use the formulaµ =XxP (x)which in practice works out as follows:x P (x) xP (x)0 0.23 01 0.38 0.382 0.22 0.443 0.13 0.394 0.03 0.125 0.01 0.05≥ 6 0 0Total 1.00 1.38giving the same answer.The value µ (Greek mu) is called the mean or expected value ofthe random variable X.7One place the calculation of µ is useful is in considering the valueof insurance policies.Example: The actuary for an insurance company determines thatyou have a 5% chance in any one year of requiring repair damageto your car, which we will simplify by saying the payout is exactly$2,000. Alternatively, with a 0.1% chance, the car will have to bereplaced completely, and the payout is $20,000. For this policy,the insurance company charges a premium of $150. Is this fair?8In this case let X be “the payout on the policy in a given year”.The possible values and their probabilities are as follows:x P (x) xP (x)0 0.949 02000 0.05 10020000 0.001 20Total 1.00 120So the expected payout per policy is $120. When allowing forthe insurance company’s operating costs and a built-in factor forrisk, the premium of $150 does not seem unreasonable.Real insurance calculations are done a lot like that, but of coursethey are based on many more possible payout scenarios and also,they take into account a lot more individual information (yourage, where you live, record of past claims, etc.)9Spread of a random variableSo far we have only talked about the mean of a random variable,which represents the most natural measure of the center of adistribution. However there are other concepts such as median,mode,...., reflecting our earlier discussion about summarizing thecenter of a sample of data.The spread of a distribution may be measured in numerous ways,of which the standard deviation is the most common. Usuallythe standard deviation is written σ (Greek sigma).We don’t discuss here the formula for computing σ. However, itwill come up when we deal with specific examples later.10Probability Distribution of a Categorical VariableIn principle, categorical variables lie outside the framework ofthis discussion since the outcomes are not numerical, but we canmake them fit by adopting certain conventions. In particular, fora categorical variable with only two outcomes, we can label them0 and 1 to turn them into a discrete numerical variable.Example. In an experiment in which rats are exposed to a toxicsubstance, 35% of them die. We can label the outcomes as 0for alive and 1 for dead. The probability distribution is then:x P (x) xP (x)0 0.65 01 0.35 0.350.35The expected value µ is equal to the probability of the outcomelabelled 1.11Continuous DistributionsA random variable is continuous if it takes a continuum of pos-sible values. For example if X is “height of a person in inches”,there’s no requirement that is be a whole number of inches.• A question like “what is the probability that X = 64?” ismeaningless because nobody has a height