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## Problem 1. This semester, 45 students have internships with one of three ?rms; ?rm A employs 20 stud

Problem 1. This semester, 45 students have internships with one of three ?rms; ?rm A employs 20 students, ?rm B employs 15 students, and ?rm C employs 10 students. Of these 45 students, six are randomly selected by a Daily Texan reporter for a story on student interns.(a) What is the probability that all six students have internships with Firm A?(b) What is the probability that all six students have internships with the same ?rm?(c) What is the probability that at least two ?rms will have interns in the sample?(d) What is the probability that all three ?rms will have interns in the sample?Problem 2. In ?ve-card poker, a ?ush consists of ?ve cards of the same suit; a straight consists of ?ve cards  of consecutive denominations (e.g., 2, 3, 4, 5, and 6) (aces can be high or low, and suit doesnât matter); and a straight ?ush is a hand which is both a straight and a ?ush (?ve consecutive cards of the same suit).Assuming a hand of ?ve cards dealt from a standard 52-card deck, what is the probability of being dealt a ?ush, a straight, and a straight ?ush?Problem 3. In class, we saw that âfalse positivesâ can be a problem when testing for a very rare event(such as a terrorist in an airport, or having an unusual disease). In this problem, we want to study exactly how common an event needs to be for a positive test to actually mean the event is likelier to have occurred than not.(a) A cheaper test for rubella combinatorica will successfully identify an infected patient 95% of the time.On the other hand, if you are uninfected, the test will still identify you as infected 5% of the time. Letp be the proportion of the population truly infected with rubella combinatorica. For what interval of pvalues will a positive test result mean the probability of actually having rubella combinatorica is greaterthan 0.5?(b) Generalize part (a) as follows: consider any test where the âfalse negativeâ rate is E ? (i.e., the probability that the test will not identify an infected patient) and the âfalse positiveâ rate is E + (the probability that the test will mistakenly identify an uninfected patient as having rubella combinatorica). What is the minimum value of p for which a positive test outcome means the probability of having rubella combinatorica is greater than 1/2? (Feel free to solve part (b) before part (a) if you want.)Problem 4. In class, we showed that the statement P (A) ? 1 can be derived from the three axioms (1)P (A) ? 0 for all events A; (2) P (S) = 1, and (3) P (A ? B) = P (A) + P (B) if A and B are mutually exclusive events. If axiom (1) were replaced with P (A) ? 1, could we derive P (A) ? 0 from this statement and axioms (2) and (3)? (If so, axioms (2), (3), and P (A) ? 1 are equivalent to (1), (2), and (3).)Problem 5. Consider the three events A, B, and C shown in the Venn diagram below, which shows theprobability of each combination of events. Are these three events mutually independent? List all of theconditions they must satisfy to be mutually independent, and indicate whether these events satisfy that condition.Problem 6. Potpourri.(a) Show that n = n?k for any integers n ? k ? 0, and furthermore explain why this result is logicalk even from the de?nition of combinations without referring to the formula.(b) Use the fundamental principle of counting to show that if there are n outcomes in a sample space, the number of events is given by 2n .(c) Under what condition does P (A | B) = P (B | A)?(d) Show that if A and B are independent events, A and B are also independent.