Chapter 7: Discrete Probability

Question: Find the probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding.

(a) 30.

(b) 36.

(c) 42.

(d) 48.

Question: Suppose that \(A\) and \(B\) are events with probabilities \(p(A) = 2/3\) and \(p(B) = 1/2\).

a) What is the largest \(p(A \cap B)\) can be? What is the smallest it can be? Give examples to show that both extremes for \(p(A \cap B)\) are possible.

b) What is the largest \(p(A \cup B)\) can be? What is the smallest it can be? Give examples to show that both extremes for \(p(A \cup B)\) are possible.

Question: A run is a maximal sequence of successes in a sequence of Bernoulli trials. For example, in the sequence S, S, S,F, S, S,F,F, S, where S represents success and F represents failure, there are three runs consisting of three successes, two successes, and one success, respectively. Let R denote the random variable on the set of sequences of n independent Bernoulli trials that counts the number of runs in this sequence. Find E(R). (Hint: Show

that R = _n j=1 Ij , where Ij = 1 if a run begins at the j th Bernoulli trial and Ij = 0 otherwise. Find E(I1) and then find E(Ij ), where 1 < j ≤ n.)

Question: What is a conditional Probability that a randomly generated bit string of length four contains at least two consecutive 0s, given that the first bit is a 1?(Assume the probabilities of a 0 and a 1 are the same.)

Question: Find the probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding

(a) 50.

(b) 52.

(c) 56.

(d) 60.

Question: Recall from Definition \(5\) in Section \(7.2\) that the events \({E_1}\;,\;{E_2}\;, \ldots ,\;{E_n}\) are mutually independent if \(p\left( {{E_{{i_1}}} \cap {E_{{i_2}}} \cap \cdots \cap {E_{{i_{\rm{m}}}}}} \right) = p\left( {{E_{{i_1}}}} \right)p\left( {{E_{{i_2}}}} \right) \cdots p\left( {{E_{{i_{\rm{m}}}}}} \right)\) whenever \({i_j}\;,\;j = 1\;,\;2\;, \ldots ,\;m\), are integers with \(1 \le {i_1} < {i_2} < \cdots < {i_m} \le n\) and \(m \ge 2\).

a) Write out the conditions required for three events \({E_1}\;,\;{E_2}\), and \({E_3}\) to be mutually independent.

b) Let \({E_1}\;,\;{E_2}\), and \({E_3}\) be the events that the first flip comes up heads, that the second flip comes up tails, and that the third flip comes up tails, respectively, when a fair coin is flipped three times. Are \({E_1}\;,\;{E_2}\), and \({E_3}\) mutually independent?

c) Let \({E_1}\;,\;{E_2}\), and \({E_3}\) be the events that the first flip comes up heads, that the third flip comes up heads, and that an even number of heads come up, respectively, when a fair coin is flipped three times. Are \({E_1}\), \({E_1}\;,\;{E_2}\), and \({E_3}\) pairwise independent? Are they mutually independent?

d) Let \({E_1}\;,\;{E_2}\), and \({E_3}\) be the events that the first flip comes up heads, that the third flip comes up heads, and that exactly one of the first flip and third flip come up heads, respectively, when a fair coin is flipped three times. Are \({E_1}\;,\;{E_2}\), and \({E_3}\) pairwise independent? Are they mutually independent?

e) How many conditions must be checked to show that \(n\) events are mutually independent?

\(X(s)\)\(X(s)\)Question: Let be a random variable, where is a nonnegative \(s \in \(X(s) \ge k\)\({A_k}\)S\)integer for all, and let be the event that . Show that \(E(X) = \sum\nolimits_{k = 1}^\infty {p({A_k})} \).

Question: Let E be the event that a randomly generated bit string of length three contains an odd number of 1s, and let F be the event that the string starts with 1. Are E and F Independent?

Question: Find the probability of selecting none of the correct six integers in a lottery, where the order in which these integers are selected does not matter, from the positive integers not exceeding

(a) 40.

(b) 48.

(c) 56.

(d) 64.

Question: Suppose that \(A\) and \(B\) are events from a sample space \(S\) such that \(p(A) \ne 0\) and \(p(B) \ne 0\). Show that if \(p(B\mid A) < p(B)\), then \(p(A\mid B) < p(A)\).