UNIT 3 LAB: Is It Fair? Student Name: ________________ Learning Outcomes: Given real world probability data or odds relating to games of chance, insurance policies, etc., calculate the expected value for a specified outcome. Use expected value to justify decisions for given scenarios. Introduction: A game is said to be fair if the expected value is $0. Another way to think of this is that each player wouldn't necessarily have the same chance of winning, but each would expect to earn equal winnings in the long run. Consider a coin toss. Suppose we play a coin toss game where you give me a dollar when the coin lands heads, and I give you a dollar when it lands tails. If we play the coin toss game many times, in the long run neither of us will come out ahead. Obviously this isn't what happens at a casino or with a lottery. Those aren't fair as the house or the state always will bring in more money than they will pay out to winners. Directions: For questions 1-8, find the expected value for each player AND show the game is fair or indicate how you could change the rules to make it fair. 1. COIN Rules: Toss three coins. (Constructing a tree diagram of the outcomes might be helpful.) Player A wins $6 if all land heads-up Player B wins $6 if all land tails-up Player C wins $1 if the coins are mixed 2. ROCK, PAPER, SCISSORS: (Constructing a tree diagram of the outcomes might be helpful.) Player A wins $15 if all players are the same Player B wins $ 3 if two players are the same Player C wins $ 9 if all players are different (Hint: There are 27 possibilities.) 3. MARSHMALLOW Rules: Toss a marshmallow onto the table. Player A wins $1 if the marshmallow lands on its end. Player B wins $1 if the marshmallow lands on its side. Fill in the table below as you perform your experiment. Then finish filling in the table after you perform the experiment to come up with the probability that a marshmallow will land on its end and the probability that a marshmallow will land on its side. Calculate the expected values and make the game fair if it is not fair. EXPERIMENT RESULTS Tally Probability (fraction) End ______/ total = ______/______ Side ______/ total = ______/______ Totals 1 4. CAROLINA PANTHERS Rules: For 2013 season the odds against Carolina Panthers winning the Super Bowl are 66:1. (http://www.oddsshark.com/nfl/super-bowl-futures) (Hint: If the odds were 5:1 for A/B, then if 6 people are voting then 5 would vote for A and 1 would vote for B.) Friend A gets $100 if the Panthers win the Super Bowl. Friend B gets $2.00 if they do not win the Super Bowl. 5. NASA Rules: The odds against an asteroid (0.5 to 1 km in diameter) hitting the earth in the next 10 years are 999:1 (http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/danger.html) Two friends make a bet and wait 10 years. Friend A gets $1000 if an asteroid has struck. Friend B gets $1 if it has not. (Hint: This needs to be EXACTLY fair.) Complete the tables below to help you with questions 6 - 7. sum of dice # of outcom es sum of dice # of outcom es 1 2 3 4 5 6 8 9 10 11 12 0 7 4 total number of outcomes 6. DICE SUM Rules: Roll 2 dice and find the sum. Player A wins $3 if the sum is < 6 Player B wins $1 if the sum is 7 Player C wins $3 if the sum is > 8 7. EVEN ODD DICE Rules: Roll 2 dice and find the sum. Player A wins $1 if the sum is odd Player B wins $1 if the sum is even 8. Make up rules for a game involving 3 coin tosses with 2 players. Show how it is fair. Directions: For questions 9-10, find the expected values for each option. Explain your answers in complete sentences using the expected value to support your answer. 9. Shawn and Maddie are thinking about purchasing a foreclosed property for $55,000 and think it will cost an additional $40,000 to fix up the property. They feel that they can resell the property for $120,000 with probability 0.15, $100,000 with probability 0.45, $80,000 with probability 0.25, and $60,000 with probability .15. Compute AND interpret the expected profit for reselling the property. Should they purchase the property? Explain your answer in a meaningful sentence using the expected value. 10. The lottery is complicated. Let's simplify it. We'll look at two different $1 scratch off tickets. Let's assume both of them just came out, and no one has bought any tickets yet. We will not consider tax either. You are trying to decide which one you should buy. Let's look at the odds. The following websites show the information they actually give the public: http://www.nc-educationlottery.org/uploads/docs/htp385.pdf http://www.nc-educationlottery.org/uploads/docs/htp394.pdf You will notice they don't tell you how many total tickets there are, but we can figure it out using ratios. Let's look at the Hot 5's. We only need to use one of the odds. Pick the most round numbers to use. total 1 of prizes for $ 5,000 12 We will set up the following odds for $ 5,000= = of tickets = 600,000 ? We are solving the following equation 1 12 = 600,000 x . So we get 7,200,000 total tickets. If we add all the total prizes in the table and then subtract that sum from the total 7,200,000 tickets, we get how many tickets do not win. There are 5,674,038 losing tickets. We'll set up a table for the HOT 5's. Essentially the probability is the odds. We will need to figure the odds for losing. We do the following calculation 5,674,038/7,200,000 which is approximately 0.788060833. If we look at the price amount $5,000, we have a profit of $5,000 - $1 = $4,999 and using the odds we have a probability of 1/600,000 = 0.000001667. Similarly, we can perform those calculations for the LUCKY 7's. We get the following after performing the calculations. We get there are 11,520,000 total tickets... therefore there are 9,282,582 losing tickets. Profit $4,999 $499 $99 $39 $19 $9 $4 $1 $0 -$1 HOT 5's Table Profit * Probability Probability (Round to the nearest cent.) 0.000001667 0.000020833 0.00025 0.001666667 0.003333333 0.006666667 0.016666667 0.086655113 0.096711799 0.788060833 Expected Value = Profit LUCKY 7's Table Profit * Probability Probability (Round to the nearest cent.) $6,999 0.000001563 $99 0.000208333 $39 0.001666667 $19 0.003333333 $14 0.003333333 $9 0.006666667 $4 0.016666667 $1 0.04 $0 0.140056022 -$1 0.805779688 Expected Value = Using the two tables above (HOT 5's Table and LUCKY 7's Table), calculate the expected value for each lottery ticket. If you were going to buy one of those 2 scratch off tickets, which one should you buy? Explain your answer in a meaningful sentence using the expected values to support your