Www.crsd.org



Name: _______________________________________________ AP Statistics Powerball ActivityIntroduction: It is not difficult for students in an introductory statistics course to compute the probabilities of winning various prizes of Powerball, including the “jackpot.” Assuming a unique jackpot winner, it is not difficult to find the expected value and the variance of the probability distribution for the dollar prize amount. In certain circumstances, the expected value is positive, which might suggest that it would be desirable to buy a Powerball ticket! However, this may require more tickets than you can afford. It’s more often not desirable. Let’s take a look!The Powerball Game: The Powerball lottery game is played in 23 states including our home state of PA. For $2, a player chooses five different numbers between 1 and 69 (inclusive). In addition, the player chooses a “powerball” number between 1 and 26 (inclusive). The player wins the jackpot if his or her five numbers match the five numbers between 1 and 69 chosen without replacement in a random drawing AND if the player’s powerball number also matches the powerball number randomly drawn between 1 and 26. The jackpot is a varying large amount of money which will add its value from one drawing to the next if there is no winner. There are also other ways to win smaller prizes as well. The jackpot prize is typically several million dollars! The powerball drawing is every Wednesday and Saturday night at 10:59 p.m. Eastern Time.The Powerball Winnings: In addition to the multi-million dollar price won by matching all 5 regular numbers and the powerball number, there are several other scenarios where one can win a lesser prize. For instance, matching the first five numbers but failing to match the powerball number still wins a prize of $1,000,000. There are a total of 9 winning scenarios; these scenarios and their associated winnings are given in the table below, where n is the number of the first five numbers that are matched. Also in the table are most common scenarios, in which the $2s are lost.ScenarionPowerball?PrizeXProbability1st 5YJackpot2nd 5N$1,000,0003rd 4Y$50,0004th 4N$1005th 3Y$1006th 3N$77th 2Y$78th 2NNone9th 1Y$410th 1NNone11th0Y$412th 0NNone1. Use the following bullets to help you calculate the probabilities:There are five numbers chosen without replacement from the set S = {1,2,…,69} and one number chosen from the set S = {1,2,…,25,26}. This gives us ways to obtain your string of numbers.Now consider the number of possible ways a player could get exactly n of the first numbers right (where ) and also get the powerball correct. There are five correct numbers, of which the player gets n, and there are 64 incorrect numbers, of which the player gets . There is only one way to get the powerball number right. This gives the formula: Finally, consider the number of possible ways a player could get exactly n of the first five numbers right (where ) but get the powerball number wrong. This gives the formula: 2. When playing the real Powerball game, more than one person can win the jackpot. Let’s assume, however, for the sake of our sanity, that only one person can win this astronomical amount. The largest jackpot amount ever for the Powerball drawing occurred on January 13th, 2016 and was estimated $1,600,000,000. Let X represent the amount you will be walking away with after you have paid $2 for a ticket and the Powerball numbers are drawn. For example, if you get 4 numbers correct and the powerball correct, you win $50,000 but you’re really walking away with $50,000 - $2 = $49,998 since you paid $2 to play. Fill in the X column in the above table. Using this jackpot value, calculate the expected value that you should expect if you bought a single ticket to play Powerball for Januray 13th, 2016. In other words, what is E(X)? What is the variability in the Powerball game? Is this what you would expect?3. Powerball’s minimum jackpot is $40,000,000. If a large jackpot is won by someone then it starts all over on the following day, would the game be “worth” playing after analyzing your expected value?4. Let’s go back to the general version of the game, not knowing the jackpot value. Let Y represent the jackpot value. What would Y have to be in order for E(X) to equal zero? In other words, you broke even. ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download