91Ó°ÊÓ

A roulette wheel consists of 38 numbers, 0 through 36 and \(00 .\) Of these, 18 numbers are red, 18 are black, and 2 are green ( 0 and 00 ). You are given \(\$ 10\) and told that you must pick one of two wagers, for an outcome based on a spin of the wheel: (1) Bet \(\$ 10\) on number 23\. If the spin results in \(23,\) you win \(\$ 350\) and also get back your \(\$ 10\) bet. If any other number comes up, you lose your \(\$ 10,\) or (2) Bet \(\$ 10\) on black. If the spin results in any one of the black numbers, you win \(\$ 10\) and also get back your \(\$ 10\) bet. If any other color comes up, you lose your \(\$ 10\). a. Without doing any calculation, which wager would you prefer? Explain why. (There is no correct answer. Peoples' choices are based on their individual preferences and risk tolerances.) b. Find the expected outcome for each wager. Which wager is better in this sense?

A random number generator is used to generate a real number between 0 and 1 , equally likely to fall anywhere in this interval of values. (For instance, \(0.3794259832 \ldots\) is a possible outcome.) a. Sketch a curve of the probability distribution of this random variable, which is the continuous version of the uniform distribution (see Exercise 6.1). b. What is the mean of this probability distribution? c. Find the probability that this random variable falls between 0.25 and 0.75 .

The juror pool for the upcoming murder trial of a celebrity actor contains the names of 100,000 individuals in the population who may be called for jury duty. The proportion of the available jurors on the population list who are Hispanic is 0.40 . A jury of size 12 is selected at random from the population list of available jurors. Let \(X=\) the number of Hispanics selected to be jurors for this jury. a. Is it reasonable to assume that \(X\) has a binomial distribution? If so, identify the values of \(n\) and \(p\). If not, explain why not. b. Find the probability that no Hispanic is selected. c. If no Hispanic is selected out of a sample of size 12 , does this cast doubt on whether the sampling was truly random? Explain.