## Mini-Quiz #5

Honor Code: You must do these problems on your own, and submit your answers no later than 11 PM on Sunday, October 28.

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As usual, please include enough decimal places in your answers for me to be able to figure out what you're doing.

The circus is still in town!

Krustopher the Clown tosses ten numbered balls out into the audience, and then asks for them to be thrown back in numerical order (ball 1, then ball 2, ...). There's (independently) an 80% chance of him catching each ball.

What is the probability that he catches at least 7 of the thrown balls? %

While Krustopher is chasing down balls, Max the Knife is throwing knives at his spinning board. (Remember? Each throw is equally likely to go into either the red, white, or blue region.)

What is the expected number of throws he'll need in order to hit all three of the colors at least once? throws

Hint: This knife-throwing problem is closely related to the promotional card-collecting game.

Bailey Barnum predicts that daily demand for hotdogs at the circus's concession stands will be normally distributed, with a mean of 10,000 and a standard deviation of 1,100. The hotdogs to be used must be defrosted and delivered from off-site before the circus opens for the day. Any left over at the end of the day are fed to the lions. Hotdogs cost her $1.50 apiece, sell for $4.50, and each hotdog fed to a lion saves her $0.85 on lion-food. (The costs of buns and condiments are negligible.) Customers who want hotdogs and can't get them from the circus stands buy from an outdoors independent vendor instead.

What is Bailey’s “critical fractile” for hotdogs? %

How many hotdogs should she have defrosted in the morning? hotdogs

Fatima runs the “Lucky 7” weight-guessing game at the side-show: For a $3 fee (which she keeps, no matter what), she tries to guess a patron’s weight. (Her guesses can include fractions of a pound, since she actually sets the center of a sliding bar to point at a location on the weight scale.) The patron then gets on the scale, and if it turns out that Fatima missed by more than 7 pounds (i.e., that the scale needle is outside of the area covered by the 14-pound-wide bar), he or she wins a stuffed elephant. The elephants cost Fatima $4.50 apiece. (She buys them wholesale.) In actuality, Fatima is good enough that her guesses are correct on average. However, her potential error in a guess is normally distributed, with standard deviation equal to 5% of the patron’s true weight.

A man (who happens to weigh 200 pounds, although Fatima, of course, doesn’t know this) pays his $3.

What is the probability that Fatima has to give him an elephant? %

What is Fatima’s expected net profit from his patronage? $

How much would a patron have to weigh in order for a wager with Fatima to be a break-even proposition, i.e., offer zero expected profit (from Fatima’s perspective, since we don’t know how much the patron might value the elephant)? pounds

Hint: You could use Solver's "equal to ... value of" feature for this.

Fatima’s daily net profit averages out to about $225, with a standard deviation of $40. Her profit varies independently from day to day, following a normal distribution. The circus has just set up in Grant Park for a six-day run.

What is the probability that she nets at least $1,400 during the run? %

Ivan the Strongman knows his limits: On any given day, the maximum amount of weight he can successfully lift is normally distributed, with a mean of 320 pounds and a standard deviation of 25 pounds. He ends his act with a single attempt at a heavy lift. If his lift is successful, he’ll be paid a bonus of $5 for every pound over 300 on the bar, but if he fails, he receives no bonus for the day.

What is his expected bonus if he attempts to lift 330 pounds? $

In order to maximize his expected bonus, what weight should he attempt to lift? pounds

Hercules the Hefty, the strongman at the Cirque de la Lune, has issued a challenge to Ivan: Each will throw a 20-pound weight as far as he can, and the one who throws furthest will be declared "Champion of Strongmen." After some practicing, Ivan estimates that the distance he can throw the weight on a single attempt is normally distributed, with an expected value of 60 feet and a standard deviation of 8 feet. A spy reports that Hercules' throwing distances vary normally, with a mean of 57 feet and a standard deviation of 6 feet.

Using 25,000 simulated repetitions of the competition, estimate the probability that Ivan will win the competition. %

Comment: Since the difference of two normally-distributed random variables is also normally distributed, you should be able to work this problem out analytically, as well.

Using 25,000 simulated repetitions of the competition, estimate the expected length of the winning throw. feet

Comment: This final problem has no analytical solution.

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