Chapter Review 543- Flip a fair coin eight times. What is the probability of getting five heads? If the coin is
biased and comes up heads only 40% of the time, what is the probability of five heads
out of eight flips? of six heads out of eight flips? - What does it mean that two events are disjoint? What does it mean that two events are
independent? Are disjoint events independent? Consider two events based on rolling a
fair die one time: The first event is getting an odd number of pips on the top face, and
the second event is rolling either a 3 or a 6. Are these two events an independent pair
of events? - State Bayes' Rule. Compute the probability for rolling a fair die twice and getting
three pips both times, knowing that the first roll does result in three pips. What does
your intuition suggest as an answer? Use Bayes' Rule to verify your intuition. - Define a random variable on the sample space that gives the sum of the values on the
top faces after rolling two fair die. Suppose the sample space of 36 pairs has a uniform
probability density defined on it. Determine P (X = x) for each value x of X.
- Define a random variable X that counts the number of tails that result from flipping
a fair coin three times. Let the sample space for flipping the coin have a uniform
probability density function. Compute the expectation or mean of X. - Choose a random number from the sample space {1, 2, 3, 4, 5}, and flip a fair coin,
resulting in either heads or tails appearing on the top face. Let both sample spaces
have a uniform probability density function defined on them. Let the random variable
X have value twice the number drawn if heads is flipped and just the value of the
number drawn if tails is flipped. The mean of the random variable is 4.5. Compute the
variance and the standard deviation of X.
8.11.3 Review Questions
- Let 2 = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3}, B = {3, 5, 6}, and C ={2, 4, 6. Describe the
following events:
(a) At least one of the events A or B occurs.
(b) Exactly one of the events A or B occurs.
(c) At least one of the events A, B, or C occurs.
(d) Exactly one of the events A, B, or C occurs.
(e) All three of the events A, B, and C occur.
(f) Exactly two of the events A, B, or C occur.
(g) At least two of the events A, B, or C occur.
(h) None of the events A, B, and C occurs.
(i) No more than one of the events A, B, or C occurs.
(j) No more than two of the events A, B, or C occur.
(k) A occurs, but neither B nor C occurs.- An electronic system of n components is said to be a series system if failure of at least
one component causes a system failure. It is called a parallel system if the system fails
only when all components fail. Suppose that 15 components are connected in a parallel
system and that each component has probability 1/20 of working properly. What can
be said about the probability of a system failure? - What is the probability that a randomly chosen integer is a member of the set of num-
bers divisible by 3? Not divisible by 5? Divisible by either 4 or 6?