Chapter 2

Let \(A\) and \(B\) be two events in a sample space for which \(\mathrm{P}(A)=2 / 3\), \(\mathrm{P}(B)=1 / 6\), and \(\mathrm{P}(A \cap B)=1 / 9\). What is \(\mathrm{P}(A \cup B) ?\)

Let \(E\) and \(F\) be two events for which one knows that the probability that at least one of them occurs is \(3 / 4\). What is the probability that neither \(E\) nor \(F\) occurs? Hint: use one of DeMorgan's laws: \(E^{c} \cap F^{c}=(E \cup F)^{c}\).

Let \(C\) and \(D\) be two events for which one knows that \(\mathrm{P}(C)=0.3, \mathrm{P}(D)=\) \(0.4\), and \(\mathrm{P}(C \cap D)=0.2\). What is \(\mathrm{P}\left(C^{C} \cap D\right) ?\)

When \(\mathrm{P}(A)=1 / 3, \mathrm{P}(B)=1 / 2\), and \(\mathrm{P}(A \cup B)=3 / 4\), what is a. \(\mathrm{P}(A \cap B)\) ? b. \(\mathrm{P}\left(A^{c} \cup B^{c}\right)\) ?

Let \(A\) and \(B\) be two events. Suppose that \(\mathrm{P}(A)=0.4, \mathrm{P}(B)=0.5\), and \(\mathrm{P}(A \cap B)=0.1\). Find the probability that \(A\) or \(B\) occurs, but not both.

Suppose the events \(D_{1}\) and \(D_{2}\) represent disasters, which are rare: \(\mathrm{P}\left(D_{1}\right) \leq 10^{-6}\) and \(\mathrm{P}\left(D_{2}\right) \leq 10^{-6}\). What can you say about the probability that at least one of the disasters occurs? What about the probability that they both occur?

We toss a coin three times. For this experiment we choose the sample space $$ \Omega=\\{H H H, T H H, H T H, H H T, T T H, T H T, H T T, T T T\\} $$ where \(T\) stands for tails and \(H\) for heads. a. Write down the set of outcomes corresponding to each of the following events: \(A\) : "we throw tails exactly two times." \(B\) : "we throw tails at least two times." \(C:\) "tails did not appear before a head appeared." \(D:\) "the first throw results in tails." b. Write down the set of outcomes corresponding to each of the following events: \(A^{c}, A \cup(C \cap D)\), and \(A \cap D^{c}\).

We toss a coin three times. For this experiment we choose the sample space $$ \Omega=\\{H H H, T H H, H T H, H H T, T T H, T H T, H T T, T T T\\} $$ where \(T\) stands for tails and \(H\) for heads. a. Write down the set of outcomes corresponding to each of the following events: \(A\) : "we throw tails exactly two times." \(B\) : "we throw tails at least two times." \(C:\) "tails did not appear before a head appeared." \(D:\) "the first throw results in tails." b. Write down the set of outcomes corresponding to each of the following events: \(A^{c}, A \cup(C \cap D)\), and \(A \cap D^{c}\).

An experiment has only two outcomes. The first has probability \(p\) to occur, the second probability \(p^{2}\). What is \(p\) ?

In some experiment first an arbitrary choice is made out of four possibilities, and then an arbitrary choice is made out of the remaining three possibilities. One way to describe this is with a product of two sample spaces \(\\{a, b, c, d\\}:\) $$ \Omega=\\{a, b, c, d\\} \times\\{a, b, c, d\\} . $$ a. Make a \(4 \times 4\) table in which you write the probabilities of the outcomes. b. Describe the event " \(c\) is one of the chosen possibilities" and determine its probability.

\(2.16 \boxplus\) Three events \(E, F\), and \(G\) cannot occur simultaneously. Further it is known that \(\mathrm{P}(E \cap F)=\mathrm{P}(F \cap G)=\mathrm{P}(E \cap G)=1 / 3\). Can you determine \(\mathrm{P}(E)\) ?

A post office has two counters where customers can buy stamps, etc. If you are interested in the number of customers in the two queues that will form for the counters, what would you take as sample space?

In a laboratory, two experiments are repeated every day of the week in different rooms until at least one is successful, the probability of success being \(p\) for each experiment. Supposing that the experiments in different rooms and on different days are performed independently of each other, what is the probability that the laboratory scores its first successful experiment on day \(n\) ?

We repeatedly toss a coin. A head has probability \(p\), and a tail probability \(1-p\) to occur, where \(0