Chapter 7: Chapter 7

A certain event has modeled probability equal to zero. This means it will never occur-right?

Find a formula for the probability of the union of three (not necessarily mutually exclusive) events \(A, B,\) and \(C\).

Exercises \(7-12\) are based on the following table, which shows the frequency of outcomes when two distinguishable coins were tossed 4,000 times and the uppermost faces were observed. $$ \begin{array}{|r|c|c|c|c|} \hline \text { Outcome } & \text { HH } & \text { HT } & \text { TH } & \text { TT } \\ \hline \text { Frequency } & 1,100 & 950 & 1,200 & 750 \\ \hline \end{array} $$ Would you judge the second coin to be fair? Give a reason for your answer.

In Exercises \(11-24\), you are given a transition matrix \(P\) and initial distribution vector \(v\). Find \((a)\) the two-step transition matrix and (b) the distribution vectors after one, two, and three steps. [HINT: See Quick Examples 3 and \(4 .]\) \(P=\left[\begin{array}{ll}.5 & .5 \\ 0 & 1\end{array}\right], v=[1\)

The "Dogs of the Dow" are the stocks listed on the Dow with the highest dividend yield. Based on the following table, which shows the top ten stocks of the "Dogs of the Dow" list for \(2015,\) based on their performance the preceding year. $$ \begin{array}{|r|c|c|c|} \hline \text { Symbol } & \text { Company } & \text { Price } & \text { Yield } \\ \hline \mathbf{T} & \text { AT\&T } & 33.59 & 5.48 \% \\ \hline \text { VZ } & \text { Verizon } & 46.78 & 4.70 \% \\ \hline \text { CVX } & \text { Chevron } & 112.18 & 3.82 \% \\ \hline \text { MCD } & \text { McDonald's } & 93.70 & 3.63 \% \\ \hline \text { PFE } & \text { Pfizer } & 31.15 & 3.60 \% \\ \hline \text { GE } & \text { General Electric } & 25.27 & 3.48 \% \\ \hline \text { MRK } & \text { Merck } & 56.79 & 3.17 \% \\ \hline \text { CAT } & \text { Caterpillar } & 91.53 & 3.06 \% \\ \hline \text { XOM } & \text { ExxonMobil } & 92.45 & 2.99 \% \\ \hline \text { KO } & \text { Coca-Cola } & 42.22 & 2.89 \% \\ \hline \end{array} $$ If you selected three of these stocks at random, what is the probability that all three of the stocks in your selection had yields of \(3.75 \%\) or more?

An experiment is given together with an event. Find the (modeled) probability of each event, assuming that the coins and dice are distinguishable and fair and that what is observed are the faces or numbers uppermost. Three coins are tossed; the result is more tails than heads.

You are given a transition matrix \(P\) and initial distribution vector \(v\). Find \((a)\) the two-step transition matrix and (b) the distribution vectors after one, two, and three steps. [HINT: See Quick Examples 3 and \(4 .]\) $P=\left[\begin{array}{ll}1 & 0 \\ .5 & .5\end{array}\right], v=\left[\begin{array}{ll}0 & 1\end{array}\right]$

Exercises \(7-12\) are based on the following table, which shows the frequency of outcomes when two distinguishable coins were tossed 4,000 times and the uppermost faces were observed. $$ \begin{array}{|r|c|c|c|c|} \hline \text { Outcome } & \text { HH } & \text { HT } & \text { TH } & \text { TT } \\ \hline \text { Frequency } & 1,100 & 950 & 1,200 & 750 \\ \hline \end{array} $$ Would you judge the first coin to be fair? Give a reason for your answer.

Describe the sample space \(S\) of the experiment, and list the elements of the given event. (Assume that the coins are distinguishable and that what is observed are the faces or numbers that face up.) A letter is chosen at random from those in the word Mozart; the letter is neither \(a\) nor \(m\)

In fall \(2014,34 \%\) of applicants with a Math SAT of 700 or more were admitted by the University of California, Los Angeles (UCLA), while \(12 \%\) with a Math SAT of less than 700 were admitted. Further, \(36 \%\) of all applicants had a Math SAT score of 700 or more. \(^{72}\) What percentage of admitted applicants had a Math SAT of 700 or more? (Round your answer to the nearest percentage point.)