91Ó°ÊÓ

State whether the two events (A and B) described are disjoint, independent, and/or complements. (It is possible that the two events fall into more than one of the three categories, or none of them.) South Africa plays Australia for the championship in the Rugby World Cup. Let \(\mathrm{A}\) be the event that Australia wins and \(\mathrm{B}\) be the event that South Africa wins. (The game cannot end in a tie.)

Each of the following statements demonstrate a common misuse of probability. Explain what is wrong with each statement: (a) Approximately \(10 \%\) of adults are left-handed. So, if we take a simple random sample of 10 adults, 1 of them will be left-handed. (b) A pitch in baseball can be called a ball or a strike or can be hit by the batter. As there are three possible outcomes, the probability of each is \(1 / 3\). (c) The probability that a die lands with a 1 face up is \(1 / 6 .\) So, since rolls of the die are independent, the probability that two consecutive rolls land with a 1 face up is \(1 / 6+1 / 6=1 / 3\). (d) The probability of surviving a heart attack is \(2.35 .\)

During the \(2015-16\) NBA season, Stephen Curry of the Golden State Warriors had a free throw shooting percentage of 0.908 . Assume that the probability Stephen Curry makes any given free throw is fixed at 0.908 , and that free throws are independent. (a) If Stephen Curry shoots two free throws, what is the probability that he makes both of them? (b) If Stephen Curry shoots two free throws, what is the probability that he misses both of them? (c) If Stephen Curry shoots two free throws, what is the probability that he makes exactly one of them?

The most common form of color blindness is an inability to distinguish red from green. However, this particular form of color blindness is much more common in men than in women (this is because the genes corresponding to the red and green receptors are located on the X-chromosome). Approximately \(7 \%\) of American men and \(0.4 \%\) of American women are red-green color-blind. \(^{5}\) (a) If an American male is selected at random, what is the probability that he is red-green color-blind? (b) If an American female is selected at random, what is the probability that she is NOT redgreen color-blind? (c) If one man and one woman are selected at random, what is the probability that neither are redgreen color-blind? (d) If one man and one woman are selected at random, what is the probability that at least one of them is red-green color-blind?

The word "free" is contained in \(4.75 \%\) of all messages, and \(3.57 \%\) of all messages both contain the word "free" and are marked as spam. (a) What is the probability that a message contains the word "free", given that it is spam? (b) What is the probability that a message is spam, given that it contains the word "free"?

Owner-Occupied Household Size Table P.11 gives the probability function for the random variable \(^{14}\) giving the household size for an owneroccupied housing unit in the US. \({ }^{15}\) (a) Verify that the sum of the probabilities is 1 (up to round-off error). (b) What is the probability that a unit has only one or two people in it? (c) What is the probability that a unit has five or more people in it? \begin{tabular}{lccccccc} \hline\(x\) & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline\(p(x)\) & 0.217 & 0.363 & 0.165 & 0.145 & 0.067 & 0.026 & 0.018 \\ \hline \end{tabular} (d) What is the probability that more than one person lives in a US owner- occupied housing unit?

Average Household Size for Owner-Occupied Units Table P.11 in Exercise P.82 gives the probability function for the random variable giving the household size for an owner-occupied housing unit in the US. (a) Find the mean household size. (b) Find the standard deviation for household size.

Average Household Size for Renter-Occupied Units Table \(\mathrm{P} .12\) in Exercise \(\mathrm{P} .83\) gives the probability function for the random variable giving the household size for a renter-occupied housing unit in the US. (a) Find the mean household size. (b) Find the standard deviation for household size.

Getting to the Finish In a certain board game participants roll a standard six-sided die and need to hit a particular value to get to the finish line exactly. For example, if Carol is three spots from the finish, only a roll of 3 will let her win; anything else and she must wait another turn to roll again. The chance of getting the number she wants on any roll is \(p=1 / 6\) and the rolls are independent of each other. We let a random variable \(X\) count the number of turns until a player gets the number needed to win. The possible values of \(X\) are \(1,2,3, \ldots\) and the probability function for any particular count is given by the formula $$ P(X=k)=p(1-p)^{k-1} $$ (a) Find the probability a player finishes on the third turn. (b) Find the probability a player takes more than three turns to finish.

In Exercises \(\mathrm{P} .95\) to \(\mathrm{P} .99,\) determine whether the process describes a binomial random variable. If it is binomial, give values for \(n\) and \(p .\) If it is not binomial, state why not. Count the number of sixes in 10 dice rolls.