mcfarland, 22sep2000 UCLA Soc. 210a, Assignment 5: Conditional Probability 1. After reading Section 2.6 of Moore and McCabe, do the following exercises beginning on page 201: 2.81, 2.83, 2.85, 2.87, 2.89, 2.91, 2.93, 2.96 2. After reading Section 4.5 of Moore and McCabe ("General Probability Rules"), do the following exercises: on page 312: 4.35 beginning on page 340: 4.55, 4.59, 4.65 beginning on page 359: 4.75, 4.77, 4.79, 4.81, 4.83, 4.101, 4.107, 4.109, 4.111 3. A lecture is attended by 100 people. An observer counts the attendees and classifies each into one of four categories, White, Black, Latino, and Asian, also noting whether each is Male or Female. Those tallies include: 16 Latinos 12 Asians 8 Latino Females 8 Black Females 36 White Males 40 Females 44 nonWhites Begin by filling in a 4x2 frequency table whose rows are the four ethnicities and whose columns are the two genders, using information provided to calculate other entries that are not directly provided. 4. (Continuation) After the lecture, there is a random drawing for door prizes, consisting of three copies of the speaker's book. Three of the 100 tickets are drawn, and the holders of the three tickets win. What is the probability that: There are both Male and Female winners? Each of the four ethnicities has a winner? All three books go to White Males? Blacks and Latinos win the same number of books (both 0 or both 1)? 5. (Continuation) Repeat problem 4, but with each winning ticket put back, thus making it possible that the same person could win twice or even all three times. Does it make much difference? 6. Find the probability that a family with four children will NOT have equal numbers of boys and girls. To simplify this calculation, the probability .49 should be rounded to .50. What assumptions are being made in your calculations? 7. Sketch how one would answer the following, but do not attempt the actual calculations. If p(female) is .5 in some remote village, what is the probability that the next 100 births will have unequal numbers of male and females? 8. Consider a simplified 2-class version of social mobility from father to son. The following table gives conditional probabilities of son's class, given father's class, and in parentheses, marginal probabilities of father's class. Son's class upper lower total Father's upper .67 .33 1.00 (60%) class lower .25 .75 1.00 (40%) total .50 .50 1.00 (100%) (8a) Turn this table around, using Bayes' rule. Or if there is insufficient information to do that, specify what additional information is required. (8b) Of the father-son pairs in which the father is lower class, in what fraction is the son upper class? (8c) Of the father-son pairs in which the son is lower class, in what fraction is the father upper class? (8d) Construct a table identical to the above one, except replacing the four probabilities, .67, .33, .25, and .75, with the values that would obtain if the other table entries were unchanged, but father's class and son's class were statistically independent. (8e) If applied to actual mobility data from a particular time and place, such hypothetical probabilities (calculated under the hypothesis of independence) would be regarded which of the ways discussed in the lecture and web page (believed, at least approximately; etc.)? 9. This exercise asks you to work through a simplified version of calculations along the lines of those by Oliver and Glick, in which the assumption is not independence, but rather the Markov assumption of one-step dependence: that Prob(class|classes of entire line of ancestors) can be replaced by Prob(class| father's class). Note that this does NOT assume that the effects of social class end after a single generation, only that those effects are limited to effects transmitted through the intervening generation(s). (9a) Consider three successive generations, and assume the probabilities in exercise 8 apply between successive generations, first for mobility between generation 1 and generation 2, and later for mobility between generation 2 and generation 3. If generation 1 had the marginal probabilities 60% and 40% as shown in the exercise 8 table, what would be the corresponding marginal probabilities for generation 2? for generation 3? (9b) Oliver and Glick went on to calculations that involved mixing marginal probabilities estimated from one group, with conditional probabilities estimated from another group, to theorize about the distinction between initial disadvantage and ongoing discrimination. You will next do something similar for this exercise. What if a different group, whose marginal probabilities were 50% and 50% (rather than 60% and 40%) in generation 1, were to experience the conditional probabilities shown in that table? What would that group's marginal probabilities be for generation 2? for generation 3? 10. A Bayesian might look at a mobility table as giving in each column the likelihood function for two hypotheses, given the observation in that column (X1 or X2). Either data outcome is possible under either hypothesis, but under hypothesis 1 the result X1 is more probable, while under hypothesis 2 the result X2 is more probable. Data X1 X2 total Hypothesis 1 .67 .33 1.00 Hypothesis 2 .25 .75 1.00 Suppose that the Bayesian had prior probabilities as follows: P(Hypothesis 1) = .8 P(Hypothesis 2) = .2 Now suppose outcome X2 was observed. How would the Bayesian revise his or her probabilities? (Calculate the posterior probabilities.) 11. Consider a second Bayesian, who might be either a regular Bayesian initially far less convinced of Hypothesis 1, or an "Empirical Bayesian" reluctant to impose subjective probabilities in the calculations. This second Bayesian uses the same likelihood function, but begins instead with a "diffuse" prior distribution: P(Hypothesis 1) = P(Hypothesis 2) = .5 Again, suppose outcome X2 was observed. How would this second Bayesian revise his or her prior probabilities, and what would be the resulting posterior probabilities? 12. If several equally competent managers (all with same value of probability of correct decision) face five crucial decisions, (p = .5) what is the probability that exactly one emerges with all five right? What is being assumed in your calculations?