Download STP 421 - Problem Set 1 1.) Suppose that three balls are sampled at

STP 421 - Problem Set 1
1.) Suppose that three balls are sampled at random and without replacement from an urn
containing 4 white balls and 2 black balls. Here, sampling at random means that each ball is
equally likely to be chosen and that the choices of different balls are independent of one another.
Sampling without replacement means that once a ball is sampled, it is permanently removed
from the urn and cannot be sampled again.
(a) Write down a sample space for this experiment assuming that we are only told the number
of white balls in the sample.
(b) Write down a sample space assuming that we also know the order in which white and/or
black balls are sampled. For example, one possible outcome would be BWB, assuming
that we first sampled a black ball, then a white ball and then again a black ball.
2.) Suppose that a pair of fair dice is rolled and that all 36 outcomes are equally likely.
(a) Calculate the probability that the dice land on a pair of numbers with a sum greater than
7.
(b) Calculate the probability that the both die land on the same number.
3.) Suppose that A and B are mutually exclusive events for which P(A) = 0.6 and P(B) = 0.3.
What is the probability that
(a) either A or B occurs?
(b) both A and B occur?
(c) neither A nor B occurs?
(d) A occurs and B does not occur?
4.) A forest contains 20 elk, of which 10 are captured, tagged and then released. Some time
later, 4 of the 20 elk are recaptured. What is the probability that exactly two of these are
tagged? What assumptions are you making?
5.) Suppose that E and F are events and let E ∩ F denote the event that both E and F occur.
Show that
P(E ∩ F ) ≥ P(E) + P(F ) − 1.
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