# STATION 1: ROULETTE. Name of Guesser Tally of Wins Tally of Losses # of Wins #1 #2

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1 Casino Lab ICM The House Always Wins! Casinos rely on the laws of probability and expected values of random variables to guarantee them profits on a daily basis. Some individuals will walk away very wealthy, while others will leave with nothing but memories. This lab is designed to allow you to analyze some of the games of chance that are typically played in casinos. (Subliminal message: keep your money in your pocket!) You must complete Stations 2 & 3 AND either Station 1 or Station 4. STATION 1: ROULETTE RULES: A roulette wheel has 38 numbered slots, 1-36, 0, and 00. Half of the slots numbered 1-36 are red and the other half are black. Both 0 and 00 are green slots. Spin the wheel with the ball inside, and make a bet of either black or red (you are not allowed to bet on green today). You win if the ball lands in the color you guessed. (Both players can make a guess (bet) on the same spin of the wheel.) SIMULATION: Play 20 games of roulette with your partner (10 each). Alternately, you can easily simulate the roulette game on your TI-83 by entering RandInt(1,38) and letting 37 = 0 and 38 = 00. Have one partner operate the calculator, and let the other person guess "red" or "black" prior to each spin. Let odd numbers represent red and evens represent black. Perform 10 simulations, and then switch jobs so you ve played 20 games. Record your results in the table below. Name of Guesser Tally of Wins Tally of Losses # of Wins #1 #2 # of Losses In what proportion of the games played did you and your partner win (totaled together)? 1. What is the theoretical probability of winning roulette by betting red? black?

2 STATION 2: BLACKJACK RULES: The game of blackjack begins by dealing 2 cards to a player, the first face-down and the second face-up on top of the first. The player has a "blackjack" or "twenty-one" if he has an Ace and a 10, Jack, Queen, or King. For simplicity s sake we will call this a win, and anything else a loss. SIMULATION: Deal 24 blackjack hands, one at a time, shuffling between each hand. That is, deal 2 cards, then check the result, then shuffle, then deal two more cards, etc. Record the number of wins (blackjacks) and losses you have. Wins: Losses: In what proportion of the games did you win? Why are you asked to shuffle between each hand? What difference does it make? 1. Find the theoretical probability of winning (i.e. getting a blackjack with 2 cards). 3. Given that the face-up card is an ace, find the probability that you have a blackjack. 4. Given that the face-up card is a 10 or face card, find the probability that you have a blackjack. 5. Let events A = face-up card is an ace and B = you get a blackjack a. Are A and B independent? Explain. b. Are A and B disjoint? Explain. 6. In the game show Catch 21 the objective is to draw cards to get to 21 as in blackjack. Would you be better off getting dealt an Ace or a 10 as the first card or does it matter? Explain.

3 STATION 3: CRAPS RULES: Roll a pair of six-sided dice. If the sum is 7 or 11, you WIN! If the sum is 2, 3, or 12, you LOSE! If the sum is any other number, that number becomes the point. Then continue rolling the dice until you either roll that point number again (WIN!) or roll a 7 (LOSE!). A new game starts after a win or loss has been recorded. SIMULATION: Play 20 games of craps with your partner. Each of you should throw the dice for 10 games. Record your results in the tables below. Game 1 st roll/point Win/Lose Game 1 st roll/point Win/Lose In what proportion of the games did you win on your first roll? In what proportion of the games did you win (in either way)? 1. What is the probability (theoretical, not based on your individual results) that you win on the first roll (obtain a sum of 7 or a sum of 11)? 3. What is the probability that you lose on the first roll (obtain a sum of 2, 3, or 12)? 4. What is the probability that you roll again after the first roll?

4 Suppose you roll an 8 on the first roll. The probability that you will win given that you rolled an 8 on the first roll is 5/11. This is because there are 5 ways to roll an 8 (win) and 6 ways to roll a 7 (lose). Since all other rolls result in rolling again, these represent all the possibilities (5 + 6 =11). 5. Find the probabilities of winning given that you roll each of the numbers by finishing and filling in the probability tree diagram for the game of craps. Win (7, or 11 on first roll) Lose (2,3, or 12 on first roll) 4 5 5/ /11 6/11 win (roll 8 again) lose (roll 7) Use the probability tree to find the probability that you win at craps. 7. It can be said that craps is the best game to play. Why is this? Does the house still have the advantage? 8. How close was your winning proportion in the games you played to the true proportion?

5 STATION 4: MONTE S DILEMMA RULES: This game is based on the old television show Let s Make A Deal, hosted by Monte Hall. At the end of each show, the contestant who had won the most money was invited to choose from among three doors: door #1, door #2, or door #3. Behind one of the three doors was a very nice prize. But behind the other two were rather undesirable prizes for example donkeys. The contestant selected a door. Then, Monte revealed what was behind one of the two doors that the contestant DIDN T pick a donkey. He then gave the contestant the option of sticking with the door she had originally selected or choosing the other door. SIMULATION: Pull an ace and two 2s from the deck of cards. These represent the 3 doors with prizes (Ace is good, 2s are bad). Partner 1 arranges the cards knowing where the ace is, and acts as game show host. Partner 2 picks a card without looking at it. Partner 1 shows Partner 2 one of the 2 s on a card he didn t pick. Partner 2 must then decide to stick with his original choice or to switch cards. The final pick of an Ace wins and a 2 loses. Perform this twenty times and record the results. Make sure to do some trials in which you switch and some in which you stick with your original choice. Modern version: Visit the web site Trial Door chosen Stick/ Switch Win/ Lose Trial Door chosen Stick/ Switch Win/ Lose In what proportion of the games did you win? In what proportion of the games in which you stayed did you win? In what proportion of the games in which you switched did you win? 1. What s the probability that you picked the door with the nice prize behind it in the first place? 2. Intuition tells us that it shouldn t make any difference whether you stick or switch. There s still a 1/3 chance that you re right. Agree or disagree? Justify by finding the probabilities of winning given that you stayed and winning given that you switched. You might find a tree diagram helpful.

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