Honor Code: You must do these problems on your own, and submit your answers no later than 11 PM on Sunday, October 7.
Name:
NetID: (three letters + three digits)
If an answer doesn't come out evenly, please give that answer to two decimal places, e.g., 23.45%.
You awaken one lovely morning to find that the circus is still in town!!! You go out to see the show.
Mack, the knife-thrower, does a routine where he throws knives at a spinning circular board. The board is divided into three equal-sized wedges, colored red, white, and blue. In actuality, Mack is competent enough that every thrown knife sticks somewhere on the board, but where (in which wedge) it hits is totally random (and the landing places of his throws are independent of one another).
What is the probability that he gets at least one knife into the red wedge in his first three throws? %
Hint: Sometimes it helps to flip an "at least" problem into an "at most" problem.
What is the probability that his first three throws all land in different-colored wedges (i.e., that he hits all three colors)? %
Hint: You might be able to "count" this problem out directly. Alternatively, you could lay it out in sequence: Something must happen, and then something else, and then something else. Of course, as a last resort you could estimate the probability via simulation.
Mack�s grand finale involves hitting apples thrown into the air by a volunteer from the audience. On a �good� day, he has a 90% chance of hitting each apple he throws at. On a �bad� day, he only hits on 50% of his throws. He�s a trouper: 75% of his days are �good� ones.
What is the probability that Mack hits the first apple? %
Hint: "It depends."
Indeed, he connects on his first throw! What�s the probability it�s a good day? %
Hint: You've "learned" something.
If he connects on his first throw, what�s the chance he�ll also connect on his second? %
Hint: Combine your approach to the first problem with your answer to the second.
Leon, the lion tamer, opens his act by trying to make each of his 7 lions jump through a flaming hoop. Just before he starts the act, his assistant informs him that four of the lions did not sleep well last night. From past experience, Leon knows that a well-rested lion will have an 85% chance of obeying his command, while a sleepy lion will have only a 65% chance of obeying.
Using 100,000 simulated observations, estimate the probability that at least 4 of Leon's 7 lions will jump through the flaming hoop. %
Hint: This isn't all that different from the "Cubs win at least 4 of their last 5 games" example we've simulated previously.
What is the margin of error in your estimate (at the 95%-confidence level)? %
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