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The game of roulette involves spinning a wheel with 38 slots: 18 red, 18 black, and 2 green. A ball is spun onto the wheel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. (a) You watch a roulette wheel spin 3 consecutive times and the ball lands on a red slot each time. What is the probability that the ball will land on a red slot on the next spin? (b) You watch a roulette wheel spin 300 consecutive times and the ball lands on a red slot each time. What is the probability that the ball will land on a red slot on the next spin? (c) Are you equally confident of your answers to parts (a) and (b)? Why or why not?

Backgammon is a board game for two players in which the playing pieces are moved according to the roll of two dice. Players win by removing all of their pieces from the board, so it is usually good to roll high numbers. You are playing backgammon with a friend and you roll two 6 s in your first roll and two 6 s in your second roll. Your friend rolls two \(3 \mathrm{~s}\) in his first roll and again in his second row. Your friend claims that you are cheating, because rolling double 6 s twice in a row is very unlikely. Using probability, show that your rolls were just as likely as his.

A Pew Research survey asked 2,373 randomly sampled registered voters their political affiliation (Republican, Democrat, or Independent) and whether or not they identify as swing voters. \(35 \%\) of respondents identified as Independent, \(23 \%\) identified as swing voters, and \(11 \%\) identified as both. \(^{21}\) (a) Are being Independent and being a swing voter disjoint, i.e. mutually exclusive? (b) Draw a Venn diagram summarizing the variables and their associated probabilities. (c) What percent of voters are Independent but not swing voters? (d) What percent of voters are Independent or swing voters? (e) What percent of voters are neither Independent nor swing voters? (f) Is the event that someone is a swing voter independent of the event that someone is a political Independent?

A Pew Research poll asked 1,306 Americans "From what you've read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?". The table below shows the distribution of responses by party and ideology, where the counts have been replaced with relative frequencies. $$\begin{array}{lcccc} &{\text { Response }} & \\ & \begin{array}{c}\text { Earth is } \\ \text { warming }\end{array} & \begin{array}{c}\text { Not } \\\\\text { warming } \end{array} & \begin{array}{c}\text { Don't Know } \\\\\text { Refuse }\end{array} & \text { Total } \\ \hline \text { Conservative Republican } & 0.11 & 0.20 & 0.02 & 0.33 \\ \text { Mod/Lib Republican } & 0.06 & 0.06 & 0.01 & 0.13 \\ \text { Mod/Cons Democrat } & 0.25 & 0.07 & 0.02 & 0.34 \\ \text { Liberal Democrat } & 0.18 & 0.01 & 0.01 & 0.20 \\ \hline \text { Total } & 0.60 & 0.34 & 0.06 & 1.00 \end{array}$$ (a) Are believing that the earth is warming and being a liberal Democrat mutually exclusive? (b) What is the probability that a randomly chosen respondent believes the earth is warming or is a liberal Democrat? (c) What is the probability that a randomly chosen respondent believes the earth is warming given that he is a liberal Democrat? (d) What is the probability that a randomly chosen respondent believes the earth is warming given that he is a conservative Republican? (e) Does it appear that whether or not a respondent believes the earth is warming is independent of their party and ideology? Explain your reasoning. (f) What is the probability that a randomly chosen respondent is a moderate/liberal Republican given that he does not believe that the earth is warming?

Assortative mating is a nonrandom mating pattern where individuals with similar genotypes and/or phenotypes mate with one another more frequently than what would be expected under a random mating pattern. Researchers studying this topic collected data on eye colors of 204 Scandinavian men and their female partners. The table below summarizes the results. For simplicity, we only include heterosexual relationships in this exercise. \(^{42}\) $$\begin{array}{lcccc} & {\text { Partner (female) }} & \\ & \text { Blue } & \text { Brown } & \text { Green } & \text { Total } \\ \hline \text { Blue } & 78 & 23 & 13 & 114 \\ \text { Brown } & 19 & 23 & 12 & 54 \\ \text { Green } & 11 & 9 & 16 & 36 \\ \hline \text { Total } & 108 & 55 & 41 & 204\end{array}$$ (a) What is the probability that a randomly chosen male respondent or his partner has blue eyes? (b) What is the probability that a randomly chosen male respondent with blue eyes has a partner with blue eyes? (c) What is the probability that a randomly chosen male respondent with brown eyes has a partner with blue eyes? What about the probability of a randomly chosen male respondent with green eyes having a partner with blue eyes? (d) Does it appear that the eye colors of male respondents and their partners are independent? Explain your reasoning.