Your television advertising campaign seems to have been very persuasive: 10,000 people who saw the ad purchased your product, while only 2,000 people purchased the produc without seeing the ad. Explain how additional data could show that your ad campaign was, in fact, unpersuasive.

Complete the following sentence. Two events \(E\) and \(F\) are mutually exclusive if their intersection is

If \(A \subseteq B\) and \(P(B) \neq 0\), why is \(P(A \mid B)=\frac{P(A)}{P(B)} ?\)

\- Construct a four-state Markov system so that both \([.5 .500]\) and \(\left[\begin{array}{ll}0 & 0.5 .5\end{array}\right]\) are steady-state vectors. HINT [Try one in which no arrows link the first two states to the last two.]

Concern the following chart, which shows the way in which a dog moves its facial muscles when torn between the drives of fight and flight. \({ }^{4}\) The "fight" drive increases from left to right; the "fight" drive increases from top to bottom. (Notice that an increase in the "fight" drive causes its upper lip to lift, while an increase in the "flight" drive draws its ears downward.) \(\nabla\) Let \(E\) be the event that the dog's flight drive is the strongest, let \(F\) be the event that the dog's flight drive is weakest, let \(G\) be the event that the dog's fight drive is the strongest, and let \(H\) be the event that the dog's fight drive is weakest. Describe the following events in terms of \(E, F, G\), and \(H\) using the symbols \(\cap, \cup\), and \(^{\prime} .\) a. The dog's flight drive is weakest and its fight drive is not weakest. b. The dog's flight drive is not strongest or its fight drive is weakest. c. Either the dog's flight drive or its fight drive fails to be strongest.

If higher and higher powers of \(P\) approach a fixed matrix \(Q\), explain why the rows of \(Q\) must be steady-state distributions vectors.