STAT 305B Fall 2007
Homework 4 (Due 9/20)
•Reading assignment: Appendix A: A.1, A.3.
•Problem 1: A bin of nuts is mixed, containing 30% 1
2inch nuts and 70% 9
16 inch nuts.
(a) Suppose a technician arbitrarily select two nuts from the bin. Describe the sample space
regarding the selected nuts. How would you expect the probabilities associated with each
outcome?
(b) Properly define a random variable associated with the random situation in (a). Is your defin-
ition a discrete random variable?
Now, there is another bin of mixed bolts with 40% 1
2inch bolts and 60% 9
16 inch bolts.
(c) Consider a technician select one nut and one bolt from the two bins respectively. Describing
the sample space in this situation. And find the probability P(A) of the event A={The nut
and the bolt are matched}.
Further, assume the total number of nuts in the bin is 100, a simple random sample of size 10 is
selected.
(d) (Optional) Describe the possibilities of the 10 selected nuts. Assume nuts of the same size
are indistinguishable from each other. Among all the possible samples, how many of them are
distinct? What if that nuts of the same size are distinguishable from each other, e.g. serial
numbers are labeled.
(e) Find the probability P(B) of the event B={10 selected nuts are all of 1
2in.}.
•Problem 2:
Let Xbe a random variable taking only three values {1,2,3}defined by the following:
P(X= 1) = 0.1, P (X= 2) = 0.2 and P(X= 3) = k.
(a) Identify the sample space, then use the Axioms of probability, find the appropriate value of k.
(b) Find the probability P(C) of C={X2−3X+ 2 = 0}.
(c) Find the probability P(D) of D={X2−3X+ 2 ≥0}.
(d) Let T=X−2.5
3, what is the probability P(E) of E={T > 0}?
•Problem 3 (Optional):
A lot contains ten pH meters, three of which are miscalibrated. A technician selects these meters
one at a time, at random without replacement, and checks their calibration.
(a) What is the probability that among the first four meters selected, exactly one is miscalibrated?
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