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STAT 224 TA : Lane Burgette Sep. 27,
DISCUSSION 3
1 Probability
- Axioms
- For any event A, P (A) ≥ 0.
- P (S) = 1
- If A 1 , A 2 , · · · , An are mutually disjoint, then
P (A 1 ∪ A 2 ∪ · · · ∪ An) =
∑^ n
i=
P (Ai)
- Conditional Probability
- For any two events A and B with P (B) > 0, the conditional probability of A given that B has occurred is defined by P (A|B) =
P (A ∩ B)
P (B)
- The Multiplication Rule P (A ∩ B) = P (A|B) · P (B)
- The Law of Total Probability Let A 1 , · · ·, Ak be mutually exclusive and exhaustive events. Then for any other event B,
P (B) = P (B|A 1 )P (A 1 ) + · · · + P (B|Ak)P (Ak )
∑^ k
i=
P (B|Ai)P (Ai)
- Bayes’ Theorem Let A 1 , · · ·, Ak be collection of k mutually exclusive and exhaustive events with P (Ai) > 0 for i = 1 , · · · , k. Then for any other event B for which P (B) > 0,
P (Aj |B) =
P (Aj ∩ B) P (B)
P (B|Aj )P (Aj ) ∑k i=1 P^ (B|Ai)^ ·^ P^ (Ai)^
, j = 1, · · · , k
- Independence
- Two events A and B are independent if P (A|B) = P (A) and are dependent otherwise.
- A and B are independent if and only if
P (A ∩ B) = P (A) · P (B)
- Events A 1 , · · · , An are mutually independent if for every k (k = 2, 3 , · · · , n) and every subset of indices i 1 , i 2 , · · · , ik, P (Ai 1 ∩ Ai 2 ∩ · · · ∩ Aik ) = P (Ai 1 ) · P (Ai 2 ) · · · · · P (Aik )
2 Exercises
2.27. An academic department with five faculty members- Anderson, Box, Cox, Cramer, and Fisher- must select two of its members to serve on a personnel review committee. Because the work will be time-consuming, no one is anxious to serve, so it is decided that the representative will be selected by putting five slips of paper in a box, mixing them, and selecting two.
a. What is the probability that both Anderson and Box will be selected?
b. What is the probability that at least one of the two members whose name begins with C is selected?
STAT 224 TA : Lane Burgette Sep. 27,
c. If the five faculty members have taught for 3, 6, 7, 10, and 14 years, respectively, at the university, what is the probability that the two chosen representatives have at least 15 years’ teaching experience at the university?
Let us consider rolling two six-sided fair dice. Let A be the event that the sum of the two rolls is 8. Let B be the event that the sum of the two rolls is 7. Let C be the event that the first roll is a 3.
a. Are the events A and C dependent, or independent?
b. Are the events B and C independent?
Let us say that we interview a group of people in a cancer hospital. Suppose that a people are smokers with lung cancer, b people are non-smokers with lung cancer, c are smokers without lung cancer, and d of the people are non-smokers with no lung cancer. Express all the relevant conditional probabilities in terms of a, b, c, and d.
2.34. A production facility employs 20 workers on the day shift, 15 workers on the swing shift, and 10 workers on the graveyard shift. A quality control consultant is to select 6 of these workers for in-depth interviews. Suppose the selection is made in such a way that any particular group of 6 workers has the same chance of being selected as does any other group (drawing 6 slips without replacement from among 45).
a. How many selections result in all 6 workers coming from the day shift? What is the probability that all 6 selected workers will be from the day shift?
b. What is the probability that all 6 selected workers will be from the same shift?
c. What is the probability that at least two different shifts will be represented among the selected workers?
d. What is the probability that at least one of the shifts will be unrepresented in the sample of workers?
2.41. A mathematics professor wishes to schedule an appointment with each of her eight teaching assistants, four men and four women, to discuss her calculus course. Suppose all possible orderings of appointments are equally likely to be selected.
a. What is the probability that at least one female assistant is among the first three with whom the professor meets?
b. What is the probability that after the first five appointments she has met with all female assistants?
2.51. One box contains six red balls and four green balls, and a second box contains seven red balls and three green balls. A ball is randomly chosen from the first box and placed in the second box. Then a ball is randomly selected from the second box and placed in the first box.
a. What is the probability that a red ball is selected from the first box and a red ball is selected from the second box?
b. At the conclusion of the selection process, what is the probability that the numbers of red and green balls in the first box are identical to the numbers at the beginning?
