Standard Level Mathematics Internal Assessment: Lottery Mathematics - Assignment Example

Lottery Mathematics Mathematics is a wide field. It entails a number of topics that range from algebra, geometry, statistics, and probability to business mathematics. These topics are a reflection of real life circumstances. For instance, probability as a topic is used in lottery games where one has to select a single number or chance out of the many provided options. Thus, this paper will discuss a number of mathematical topics with examples that match their application in real life situations. To start with, lottery mathematics is an application of probability as a mathematics topic. The game uses numbers and mathematical computations to determine the chances of an individual winning. The application of probability in lottery was started by K. J. Nurmela and P. R. J. Östergård. All the characters that were involved in this great invention were lotto designers. Lottery has handed the community an application of probability to the social aspect which is gaming (Nurmela & Ostergard, 22). The lottery game is based on similarity of numbers drawn to those on the ticket; with that probability of this similarity is a large number so it is almost practical that the game is won out of luck and not experience There are a number of games or rather lotteries that have existed across these ages. For instance, power ball is a lottery game that has been played for so many years. Many lotteries use a Powerball or rather a bonus ball. Thus, Powerball refers to an American game which is sold by exactly forty five lotteries (45) as a jackpot game. The entire game is coordinated by non-profit organizations which are formed with a sole purpose of facilitating the lotteries. There is no any maximum limit for the Powerball lottery, the ball will just roll until all the figures or numbers get matched. This game uses 5 out of 59 white balls and 1 out of 35 Powerballs matrix in which the numbers that are winning are chosen from. Powerball is played with a minimum bet of two dollars. In every game, the players select 5 numbers from the given set of 59 balls (white) and 1 number from the 35 red balls. Thus, the number chosen from the red balls might be the same as the number chosen from the white balls. In addition, the players are allowed to select their own numbers or the numbers are selected from specific terminals referred to easy pick. In every drawing, the number that wins is selected using the two ball equipment; one contains the red balls and the other the white balls. Thus, for one to win, the 5 number of white balls are drawn from the equipment followed by the one number of the Powerball from the second machine. Therefore, the games that match at least 3 white balls or the red balls win the game. Thus, Lottery game applies probability in determination of the winner. For example, in a game that involves careful selection of 5 numbers from 40 and winning is determined when the 5 numbers match the other 5 picked from the same pool of numbers. In this game, where 5 characters are to be chosen from 40, five numbers are picked from a draw that contains numbers within a range of 40. In situations where the numbers drawn are similar to the numbers on the ticket the bearer of the ticket emerges the winner and the order of appearance of the numbers on the ticket is not considered. In the above example, the first draw involves picking one number in a total of 40 differently numbered lottery balls, thus a probability of 1 in 40, the second ball is picked from 39 balls so there are chances of predicting the is 1 in 39. In each of the 40 possible ways of choosing the first ball, there are 39 different possible ways of picking the second ball from the bag. On picking the third number, there are 38 possible ways of predicting the number. The chances of picking the third number from 40 is 1 in 40 × 39 × 38 and the probability of picking the second from 49 is 1 in 40 × 439. This trend is continuous, thus having the probability of picking the fourth number as 1 in 40 × 39 × 38 × 37, picking the fifth ball as 1 in 40 × 39 × 38× 37× 36. The other significant aspect is the fact that the order in which the numbers appear does not matter. So, the five numbers can appear as 1, 2, 3, 4, 5 or 5, 4, 3, 2, 1. In general the numbers can be drawn in 5 × 4 × 3 × 2 × 1= 120 different ways. This is basically called combination function which involves the division written as 40! / (5! × (40 - 5)!). It is never a case in lottery to make assumptions that the ball will be drawn in a certain order because matching of the numbers is not based on the order that the numbers appear but on the similarity of the numbers drawn with those on the ticket. Another example is of a lottery game involving 32 from which 8 numbers are to be picked in which ever order. Thus, the probability that the numbers picked match the numbers on the ticket can be calculated as follows: (32 ) * ( 32) * (32 ) *( 32) * ( 32 ) * ( 32 ) * ( 32) =109951162776 8 7 6 5 4 3 2 40320 This therefore implies that there are 1 in 27,269,633 chances of matching all 8 numbers, this kind of lottery can be won by not necessarily having the right combination but basically matching more than anybody else. Lottery has come a long way to benefit the lives of people across the globe. As much as these games are fifty – fifty; whether a win or lose, people become rich day after day. Thus, it is not only a chance to win, but it is also an opportunity to easily earn from a bet probability. One major reason as to why I like lottery mathematics is that it is an easy way to invest and earn or obtain money that can people take years to earn. Hence, this bit of mathematics in lottery (probability) has enhanced or rather played a role in developing my positive attitude towards the topic (Barboianu, 5). For instance, I once participated in a lottery game whereby a winner was expected to have a probability of less than 5 matching numbers. How these computations were performed caught my eyes; the use of probability distribution, combination and permutation. I learnt that permutations uses or rather considers the order whereas combinations do not. Thus, in the lottery tickets, permutations were only used if the number of in the participant’s tickets matched with the order of the winning draw. In my entire participation, these mathematical aspects never got erased from my mind. Thus, they developed my interest in learning and reading more probability at school. Moreover, after emerging the winner of $500 in that lottery, I could stop taking part in as many lotteries as possible since using probability made me assume that the more I compete, the high my chances of winning. The following formulas were used in the lottery game I participated in were: P(n,k) = n!/(n-k)! for permutations that never considered the order and C(n,k) = nCk = n!/ k!(n-k)! for combinations where the order is not important As people play the lottery games, the big question always lie in the chances that a ticket someone is holding will eventually win. This kind of question is an application of probability. Thus, with millions of dollars circulating in the competition, it would be wiser to have a clear perception of what transpires in lottery games (Catlin, 65). The answer to this question is the answer to a question such as, “How does Lottery work?” In lotteries, people buy tickets which have choices of different numbers from a given range. At a given time, the organizers of these lotteries generate numbers randomly using the same range as the numbers in the tickets. Thus, the grand prizes are always awarded for matching all the numbers in the system. However, in some lotteries, like the one I participated in, less amounts are offered for matching numbers and missing others. These are what I deem to be the generalities of a lottery. Many people purchase tickets not knowing the kind of game they are engaging themselves in. Therefore, it would be better for those who participate in lotteries to have a prior idea of how a win will come by or rather what happen for them to win prizes (Haigh, 111). Lottery, however, as a game it has one major challenge to the players. It is extremely difficult for an individual to determine the total number of tickets. Many people tend to use mathematics or rather probability to estimate or calculate the how many tickets are needed for a win. Thus, in this game, there is no guarantee or a mathematical means through which the numbers on the ticket can be obtained prior to the game. Work Cited Barboianu, Catalin. The Mathematics of Lottery: Odds, Combinations, Systems. INFAROM Publishing. 2009. Print Catlin, Don. The Lottery Book: The Truth Behind the Numbers. Bonus Books. 2003. Print Haigh, John. Taking Chances. Winning with Probability. Oxford University Press. 2003. Print Nurmela, Kari J. & Ostergard, Patric R. J. Constructing Covering designs by Simulated Annealing. Helsinki University, Digital Systems Laboratory. 1993. Print Read More

Thus, for one to win, the 5 number of white balls are drawn from the equipment followed by the one number of the Powerball from the second machine. Therefore, the games that match at least 3 white balls or the red balls win the game. Thus, Lottery game applies probability in determination of the winner. For example, in a game that involves careful selection of 5 numbers from 40 and winning is determined when the 5 numbers match the other 5 picked from the same pool of numbers. In this game, where 5 characters are to be chosen from 40, five numbers are picked from a draw that contains numbers within a range of 40.

In situations where the numbers drawn are similar to the numbers on the ticket the bearer of the ticket emerges the winner and the order of appearance of the numbers on the ticket is not considered. In the above example, the first draw involves picking one number in a total of 40 differently numbered lottery balls, thus a probability of 1 in 40, the second ball is picked from 39 balls so there are chances of predicting the is 1 in 39. In each of the 40 possible ways of choosing the first ball, there are 39 different possible ways of picking the second ball from the bag.

On picking the third number, there are 38 possible ways of predicting the number. The chances of picking the third number from 40 is 1 in 40 × 39 × 38 and the probability of picking the second from 49 is 1 in 40 × 439. This trend is continuous, thus having the probability of picking the fourth number as 1 in 40 × 39 × 38 × 37, picking the fifth ball as 1 in 40 × 39 × 38× 37× 36. The other significant aspect is the fact that the order in which the numbers appear does not matter. So, the five numbers can appear as 1, 2, 3, 4, 5 or 5, 4, 3, 2, 1.

In general the numbers can be drawn in 5 × 4 × 3 × 2 × 1= 120 different ways. This is basically called combination function which involves the division written as 40! / (5! × (40 - 5)!). It is never a case in lottery to make assumptions that the ball will be drawn in a certain order because matching of the numbers is not based on the order that the numbers appear but on the similarity of the numbers drawn with those on the ticket. Another example is of a lottery game involving 32 from which 8 numbers are to be picked in which ever order.

Thus, the probability that the numbers picked match the numbers on the ticket can be calculated as follows: (32 ) * ( 32) * (32 ) *( 32) * ( 32 ) * ( 32 ) * ( 32) =109951162776 8 7 6 5 4 3 2 40320 This therefore implies that there are 1 in 27,269,633 chances of matching all 8 numbers, this kind of lottery can be won by not necessarily having the right combination but basically matching more than anybody else.

Lottery has come a long way to benefit the lives of people across the globe. As much as these games are fifty – fifty; whether a win or lose, people become rich day after day. Thus, it is not only a chance to win, but it is also an opportunity to easily earn from a bet probability. One major reason as to why I like lottery mathematics is that it is an easy way to invest and earn or obtain money that can people take years to earn. Hence, this bit of mathematics in lottery (probability) has enhanced or rather played a role in developing my positive attitude towards the topic (Barboianu, 5).

For instance, I once participated in a lottery game whereby a winner was expected to have a probability of less than 5 matching numbers. How these computations were performed caught my eyes; the use of probability distribution, combination and permutation. I learnt that permutations uses or rather considers the order whereas combinations do not. Thus, in the lottery tickets, permutations were only used if the number of in the participant’s tickets matched with the order of the winning draw.

In my entire participation, these mathematical aspects never got erased from my mind.

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