The Birthday Paradox - Assignment Example

The Birthday Paradox Introduction The birthday paradox is an interesting concept in mathematics and it has much of statistical methods to be implied. In this paper, the attempt is to imply birthday paradox to a football team where there are 23 players. If there are 23 people in a football team then there is an issue when it comes to their birthday. Basically, in a foot ball team with 23 people there is almost a 50 % chance for the football team players to share their birthday. In a foot ball team with 23 people, there is 50 – 50 chance for two people to have birthday on the same date. On the other hand, if there are 75 people in a foot ball team there is 99.9% chance for two people to share birthday at same time. Frankly speaking, birthday paradox is a true and strange calculation method which is of counter intuitive nature. From a deeper angle, “paradox” is a brain game as our brains cannot handle the compounding power of exponents naturally. Here, the expectation is to consider probabilities to be linear and expect scenarios to be faulty assumptions. According to Fletcher (2014) “ In its most famous formulation, the birthday paradox says that you only need a group of 23 people for there to be a greater than 50% chance that two of them share the same birthday”. Generally speaking, there is a possibility for the people to think that having 365 days in a year intuition can be applied to the birthday paradox. This means the intuitive answer which comes out of the situation would be basically very small. But it is important to comprehend that intuition works faulty and thus birthday paradox arises. The birthday paradox is a great mathematical concept and also it is one of the greatest hits in mathematical world. In its formulation, birthday paradox indicates that it is necessary to have 23 people in a group in order for the birthday paradox to apply. It is said that birthday paradox is not a logical paradox and it happens on an unexpected basis. Paradox Problem The paradox problem can be understood with the help of coin flipping phenomenon. In this case, the issue is all about flipping a coin and looking at the chance of getting 10 heads in a row. Since the brain is untrained the thinking of a person can take a particular stand. The assumption would be that getting one head would be taken as getting 50 % chance. In case of two heads which is harder to happen the assumption would be 25% chance. In case of getting 10 heads which is even harder the assumption would be 50%, 10% or 5 %. The birthday problem is all about finding the probability that also in relevance to a group. Basically, in the birthday problem there exist at least one pair of people who share birthday on the same day and months and this is definitely an assumption. According to Scientific American (2012 ) “There are multiple reasons why this seems like a paradox. One is that when in a room with 22 other people, if a person compares his or her birthday with the birthdays of the other people it would make for only 22 comparisons—only 22 chances for people to share the same birthday”. The intention of the birthday problem is to find the probability when there is a group of N people. This means there is a fixed person who is sharing birthday with any other remaining N- 1 people in the foot ball team. Understanding the Problem Understand the birthday paradox problem means comprehending what is the real concept in it. Here, the effort is to imply statistical calculation in relevance to the birthday of 23 people in a football team. In the football team of 23 people ,comparing the birthday of first person in the team with others gives an opportunity to match birthday with of the first person with 22 other team members. In the same way, the second person in the group allows only 21 chances for a matching birthday with the rest in the team. But the second person’s chance to meet the birthday date with the first person remains one and this has been already counted hence the 22 chances of the first person to meet the birthday date with other remains 22 itself. For the same reason, the 22 chances will not be duplicated and the third person has only 20 chances. So, the total chances can be counted as 22+21+20+…+1 = 253. Hence forth, we can understand that comparing every person to rest of the team members can allow 253 different chances. Moreover, in a group of 23 individuals there are (23/2) = 23.22/2 = 253 different combinations of pairing. The Algebraic Approach When looking at birthday paradox from an algebraic angle, it is necessary to consider the probability that no two people out of a football team will have matching birthdays. It can be suggested that an arbitrary person’s birthday will be different from the second person’s birthday and the third person’s birthday is different from that of the first two team members. To comprehend better, the birthday paradox can be seen from a calculative dimension. Let us calculate the first few probabilities with regard to the birthday paradox. Here the probability is that the two persons who are strangers would not be sharing a birthday which is 364 days divided by 365 days considering that neither of them would have been born in a leap year. So we can understand that 1-364 days divided by 365 days equals to 0.00274. The chance for three team members not to share birthday can be 364 days multiplied by 363 days and then divided by 365 days or 0.991796 along with a compliment . Basically, in this scenario, a pair of matching birthday can come in to existence and algebraic equation can go into a formula like 1-364 x 363 = 0.008204 . In the same way, when considering the team of four players wherein there is a presence of matching pair then the probability can be calculated with the following algebraic equation; 1- 364 x 363 x362 = 0.016356. 365 Now, it is necessary as a part of calculation to introduce a formula that can calculate the probability of the matching pair of birthday .The formula will be considering random birthdays and this would be depicted as r. So the formula will be as follows; P match (r) = 1 – 365 Pr Here r is the random number when it comes to birthday. The result is calculated with few values of r and this could be lower or larger. Here, r the random value can be 39 or 40. Here probabilities are calculated and scatter plot is used to analyze and represent the result. The results can be depicted as follows; The Graphical Approach Another approach to the birthday paradox is the graphical representation to it. In this case, a table and a graph will be used to graph the probability and chances. According to Aldag (2007, pg.5 ) “To compute the probability that two randomly selected people have the same birthday, we note that since their birthdays are independent of each other, the two probabilities can be calculated separately and multiplied together”. To find the probability getting a match for the people in the foot ball team is essential. For this purpose, the sequential mode is used with enter n Min = 1 formula .The formula can be further explained as below; u(n) = 1 –(1-u(n-1))(366-n) and u(nMin) = (0) Here u(n) = 1 - (365 nPr n)/365^n leads to n = 40, Plot 1 Plot 2 Plot 3 n min =1 \u(n) = 1-(1-u(n-1 ))*(366-n)/365 u(n min)={0} \v(n)= v(n min)= \w(n) n u (n) 1 0 2 0.00274 3 0.0082 4 0.01636 5 0.02714 6 0.04046 7 0.05624 n = 1 n u (n) 19 0.37912 20 0.41144 21 0.44369 22 0.4757 23 0.5073 24 0.53834 25 0.5687 n = 19 n Min 1 Plot Step= 1 n Max = 50 x min = 0 Plot Start = 1 x max = 50 Plot Step= 1 xscl = 0 x min = 0 y min = 0 x max = 50 y max = 1 xscl = 0 yscl = 0 n u (n) Average 19 0.37912 0.1675 20 0.41144 0.31945 21 0.44369 0.23091 22 0.4757 0.5637 23 0.5073 0.62762 24 0.53834 0.87125 25 0.5687 0.1782 n = 19 In the birthday paradox, it is essential to depict the statistical data in graphical form. For the same purpose the calculations have been presented in the form of scatter graph. For the graph purpose, it was essential to take the mean and average in the sequential model. The graph can be seen as below; Fig 1 : Probability Graph Fig 2 : Probability Graph Conclusion Birthday paradox is an essential method in mathematical world and for the same reason it has its particular methodology. The birthday paradox which takes in to consideration number of people in a group asks that two people have birthday within same day of a particular year. However, there is a contradiction to this fact from mathematician Abramson and Moser who suggest that 14 people would suffice in such a paradox. This is not agreed by many and that is the reason why birthday paradox has a clichéd 23 has group member. The conclusion of the birthday paradox helps to reduce the complexity of statistical assumption related with it. Birthday paradox gives an insight into the comparison problem and allows in eradicating the intuition attached with the paradox. It is interesting to understand that there is a parallel mathematical method which coincides with the birthday paradox and it is partition problem. However, this mathematical method has some drawbacks which makes it a lesser important method in comparison to birthday paradox. The highlight of the birthday problem is that it gives alternative formulation and asks for average number of people needed in order to fins a pair with same birthday. This means the probability function comes in to action and thus the birthday paradox can be conducted with more strength. The birthday problem has a method of its own and in it no two people are selected in advance. The contrast of probability that someone in the foot ball team might share the same birthday is the distinctiveness of birthday paradox. References Aldag, S,, 2007. A Monte Carlo Simulation of the Birthday Problem. University of Nebraska, 7/12, 5-16. Fletcher,J. 2014. The birthday paradox at the World Cup. [ONLINE] Available at: http://www.bbc.com/news/magazine-27835311. [Accessed 30 August 15]. Scientific American. 2012. Probability and the Birthday Paradox . [ONLINE] Available at: http://www.scientificamerican.com/article/bring-science-home-probability-birthday-paradox/. [Accessed 30 September 15]. Read More