4120 Worksheet 1: Introduction Name: 1. You are at a party chatting with some guestsMr. E, Miss Take, and Sir Pwho are each holding their own pair of red and blue jars. Each jar contains all 50 commemorative state quarters. The guests take a single quarter at random from each of their jars and ips them. (a) Let A be the event that the guest ipped two heads if no information was given. Find PM). In reality, each guest tells you a single cryptic fact about the result of their two ips. let B be the extra information given. Find P(A | B) for each guest. (b) Mr. E: \"My coin from the red jar landed heads.\" (e) Miss Take: \"At least one of my coins landed heads.\" (d) Sir Prize: \"At least one of my coins is a New York quarter that landed heads.\" 2. STOP! Don't start on this question until Matt says so. (a) Navigate to this page and do the Kruskal Count magic trick for yourself. Did the trick work for you? (b) Why do you think it tends to work? (c) Suppose you pick card 1 and your friend picks card 2. Let A be the event that you both are on the same card after 52 cards have been flipped. What do you think PM) is approximately equal to? (d) What is a difculty with exactly computing P(A)? (e) Write down one thing you learned when Matt explained the results from this paper. 3. Roulette is a casino game in which a ball is thrown into a slotted, spinning wheel that has 18 red, 18 black, and 2 green numbered slots. In the standard $1-bet a gambler wins $1 if the ball lands on a black number and loses $1 if the ball lands on a green of red number. (a) What is the probability of winning $1 on a standard bet? (b) 1We'll talk about this more formally later, but the expected value of the amount won/loss W is given by E[W] = 2: (value) x (Probability of value) = ZwP(W = m). values Find the expected value of the standard bet. (c) You walk into a casino with the intention of making $1 stande bets at roulette. Your strategy to win $1 is to ist bet a dollar. If the bet wins, you walk away. If you lose the rst round, then you double your bet and bet $2. So a win will cover your $1 lost from the previous round. You proceed in this fashion, so the kth bet is for 2\" dollars, which if you lose, you proceed to bet $2\"1 in the next bet. The rst time you win, you walk away. Suppose that W is the amount you win/lose in this strategy. What is EW? (d) Hold on! Although the previous part might sound appealing, what is the issue i.e. why are casinos ne with gamblers using this strategy? (e) How much money is needed to make for a gambler to perform 1:. attempts of the strategy? (f) What is the probability of the strategy failing after n attempts? (g) Let Wn be the amount won/loss assuming you have enough money to make n attempts at the strategy. Write a formula for EW,. Don't simplify. (h) What happens as n - co in the formula you wrote down? (i) Enjoy the following table and plot: n bankroll EW n 0 0 0.0 5 32 -0.29 10 1023 -0.67 15 32768 -1.16 20 1048575 -1.79 25 33554431 -2.61 30 1073741823 -3.66 35 34359738367 -5.02 40 1099511627775 -6.78 45 35184372088831 (~35 trillion) -9.06 20 80 100 -50 -100 -150Might not get to these... 4. (From Actuary Exam P Sample Problems) The probability that a visit to a primary care physicians (PCP) office results in neither lab work nor referral to a specialist is 35%. Of those coming to a PCPs office, 30% are referred to specialists and 40% require lab work. Calculate the probability that a visit to a PCPs office results in both lab work and referral to a specialist. Hint: P(A) = 1 - P(A ) and P(A U B) = P(A) + P(B) - P(An B). 5. (Pretty tricky: From Exam P Sample Problems) You are given P(A U B) = 0.7 and P(A U B ) = 0.9. Calculate P(A)