Probability of Lottery

Published on August 2016 | Categories: Types, School Work, Essays & Theses | Downloads: 57 | Comments: 0 | Views: 361
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A brief explanation of a suprising result - that the odds of a person winning the lottery twice in their lifetime is actually very likely

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What are the odds that somebody, out of all the people in the UK, will win the lottery more than once?
Before answering the question, there are a few things which are needed to be known: • To find the probability of the number of wins in the lottery out of a certain numbers of attempts, we need to use the binomial distribution: this states that the probability of getting exactly n wins out of k attempts is given by the following formula: k! pn (1− p)n −k where p is the probability of one win.
n ! (k −n)!

• The probability of winning the national lottery here in the UK is 1 in 13,983,816 – by this, it means that the probability of winning the national lottery by putting on one ticket for a single draw is 1 in 13,983,816. The odds of winning by putting on two tickets for a single draw would be double this (ie, 2 in 13,983,816), but the odds of winning by putting on one ticket for two separate draws would not be the same – this is because the draws are independent of each other (the outcome of the first draw has no effect on the outcome of the second draw) With this now clear, we can start answering the question. Let us firstly look at the odds of an individual winning the lottery more than once. There are several factors which will have an effect on the probability of an individual winning the lottery to begin with, these being: • How often does the individual play the lottery? (eg, twice a week (for both draws), only once a week, rarely etc.) • How many tickets will the individual put on for a single draw? For the sake of ease, let us assume that the average person will put on two tickets twice a week, and that the average person will do this for about 40 years in total (we will call this a lifetimes worth of playing). Given that the average spend on the lottery is approximately £3 a week according to Camelot (the company which runs the lottery), the assumption of £4 is not too great of a leap to make, and should not effect the results by too much. Therefore, in our modelling assumptions, for the formula given in the prior

page, we are trying to find the probability that an individual will win more than once where the probability p of one win will be 2 in 13,983,816, and we want to find the probability of n wins out of 4160 draws of the lottery the individual enters. Because the probability of 0 wins out of 4160 + 1 win out of 4160 + 2 wins out of 4160 and so on adds up to 1, we can say that the probability of an individual winning more than once is equal to 1 – the probability of winning 0 times out of 4160 – the probability of winning 1 time out of 4160. Therefore, the odds of an individual winning more than once is:
0 4160 4160 ! 2 2 ×( ) ×(1− ) 4160! 0 ! 13,983,816 13,983,816 (*) 1 4159 4160! 2 2 − ×( ) ×(1− ) 4159 ! 1 ! 13,983,816 13,983,816 =1.7688404841562643144940016324297180972388006645 ...× 10−7

1−

As you can see, these aren't very good odds. However, now we need to consider the odds that any person can win the lottery more than once – this means the probability it could be you, your neighbour, some person off Jeremy Kyle, or any other person who plays the lottery. To calculate this, we can again use the binomial distribution, but instead using it to find out the probability that n people out of the total number of people who play the lottery will win the lottery more than once. As according to Camelot 70% of the adult population play the lottery in the UK, and the adult population of the UK is approximately 50 million, there are approximately 35 million people who play the lottery in the UK. Referencing back to the original question, we are considering the odds that at least one person out of 35 million people will win the lottery more than once. When phrased like this, it sounds a lot more plausible that this is a likely event. Therefore, using the binomial distribution again only with k=35 million, p equal to the value in (*) and finding the probability that at least one person will win the lottery more than once in a lifetime by subtracting the probability no-one will win the lottery more than once in a lifetime from 1, we get:
1−(1−(1.76884... ×10−7 ))35 million =0.9975...≈99.75 %

which means it is nearly almost certain there will be at least one person who wins the lottery more than once. I would try and calculate the time it takes for the odds to become fifty-fifty, but the numbers involved are too small.

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