Winning the LotteryIntroduction: The lottery was first invented as a way for the government to generate revenue. Over time, the purpose of the lottery evolved into a form of gambling that involves the drawing of numbers to win a prize. In recent years, people around the world take the chance to participate in the lottery in hopes of becoming a millionaire. I have been around friends and family, who have heard the news that people around the world or themselves have won a prize from the lottery and how it is only a game of chance to have matching numbers with the winning ticket. Therefore, the idea of winning the lottery fascinated me and initiated my curiosity to ask the question what is the chance of winning the lottery out of the many possible combinations of numbers to match the winning ticket. The purpose of this investigation is to calculate the lottery probability for six matching numbers and the lottery probability with less than six matching numbers using combinations and permutations and the statistics of winning the lottery.Rules of the Lottery: In the typical lottery of 6/49, the player chooses six non-duplicate numbers from a range of 1 to 49. To win the jackpot prize, the player needs to match with the numbers of the winning ticket. The number that is drawn first has a 1 in 49 chance of matching the winning match. When the second the number is drawn, it has a 1 in 48 chance of matching, since no number can be duplicated. Lottery Probability: Finding the lottery probability involves the use of combinations and permutations. In permutations, order is considered to be important, while in combinations, order does not matter. Depending on the type of lottery, permutations or combinations may be used. P(n,k)=n!(n-k)!C(n,k)=n!k! (n-k)!n = total number of objects , k = number of objects taken at a timeGiven that the typical lottery game of 6/49 is a combination, I will use the combination formula and substitute n = 49 and k = 6. C(49,6)=49!6! (49-6!)=49!6! * 43!=13,983,816This represents the total number of possible combinations for any 6-digit number to win the lottery. To find the probability of one person winning, I divide the 1 by 13,983,816 which equals 0.0000000715 or 0.00000715% chance of winning the lottery. If the lottery game were a permutation, I will use the permutations formula, as the order of numbers is important. P(49,6)=49!(49-6)!=49!43!=10,068,347,520 When I divide 1 by the total possible permutations of any 6-digit number, the probability of winning the lottery is 0.00000000993%. By comparing the two probabilities, it shows that the probability of winning the game where order matters is always less than or equal to the probability of winning the game in which order does not matter. Thus, this shows how there is a higher risk for games where order matters and thus, the reward would be higher. Players are also able to win prizes when they have less than six matching numbers. For a score of x, C(6, x), describes the odds of selecting x winning numbers out of six of the possible winning numbers. Therefore, there are 6-x losing numbers. Because there are six winning numbers, there are 49-6=43 numbers to choose from the losing numbers. Therefore, the number of ways for choosing from the losing numbers is C(43,6-x). Thus, to calculate the probability of winning the lottery with x matching numbers out of the six possible winning numbers, I multiplied the number of ways to choose x winning numbers out of 6 with the number of ways to choose the remaining losing numbers and divided by the the total number of possibilities to win with all six matching numbers. Probability of winning with x matching numbers =C(k, x) * C(n-k, k-x)C(n,k)n = total number of objects k = total number of winning numbers x = total number of numbers matching the winning numbersThis generalization is also known as the hypergeometric distribution, which is used to calculate probabilities when sampling without replacement. I will find the probabilities of having 0, 1, 2, 3, 4, 5, and 6 matches with the winning numbers by plugging the number of matches into the formula as x. P(x) =C(6, x) * C(43, 6-x)C(49, 6)x matchesCalculationProbabilityOdds (1/Probability)0 P(0) =C(6, 0) * C(43, 6-0)C(49, 6)0.43596 2.29381P(1) =C(6, 1) * C(43, 6-1)C(49, 6)0.413022.42122P(2) =C(6, 2) * C(43, 6-2)C(49, 6)0.132387.55413P(3) =C(6, 3) * C(43, 6-3)C(49, 6)0.0176556.65604P(4) =C(6, 4) * C(43, 6-4)C(49, 6)0.00096861032.39695P(5) =C(6, 5) * C(43, 6-5)C(49, 6)0.0000184554200.83726P(6) =C(6, 6) * C(43, 6-6)C(49, 6)0.0000000715113983816Statistics:Data on the amount of lotteries won in the past 10 yearsPlot a cumulative frequency histogram and graph of how frequent certain numbers are chosen Conclusion: Summarize findings and the significance of the resultsRelate it back to personal engagement and the real-world Sources:Dezalyx. “How to Calculate Lottery Probability.” Owlcation, Owlcation, 13 Dec. 2016, owlcation.com/stem/How-to-Calculate-Lottery-Probability.”Lottery Mathematics.” Wikipedia, Wikimedia Foundation, 6 Jan. 2018, en.wikipedia.org/wiki/Lottery_mathematics.Lane, David M. “Hypergeometric Distribution.” Hypergeometric Distribution, onlinestatbook.com/2/probability/hypergeometric.html.Lendlsmith, Jerry Jay. “How to Win the Lottery According to Math.” LottoMetrix, LottoMetrix, 7 Dec. 2017, lottometrix.com/blog/how-to-win-the-lottery-according-to-math/.”Lotto Statistics .” Lotto Numbers, World Lottery Statistics and Analysis, www.lottonumbers.com/statistics.asp.National-Lottery.com, Lottery Company Ltd. “Lotto Statistics.” UK National Lottery, www.national-lottery.com/lotto/statistics.