UK Lottery Probability

Probabilistic properties of the e-lottery system for playing the UK Lotto Summary This report focuses on the probabilistic properties of the e-lottery system for buying tickets on the UK National Lottery game, Lotto. The system is compared to buying a single ticket and 44 random distinct tickets. The probabilities of winning the various prizes are 44 times higher than for a single ticket and exactly the same as if 44 random distinct tickets were purchased. Calculation of probabilities for e-lottery syndicate Now that we have calculated the probabilities for a single ticket, let us compare them to the corresponding winning probabilities for a member of the e-lottery syndicate. Notation In order to aid mathematical derivation of the probabilistic properties of a syndicate group, it is useful to define some notation. Each syndicate group consists of 44 tickets such that every number 1-49 appears at least once. These tickets can be thought of as consisting of two types of numbers: primary and secondary. The primary numbers appear on every one of the 44 tickets. The secondary numbers appear just once. For example, here is a set of tickets for one syndicate group: The numbers are primary numbers whilst the other, , are the secondary numbers. We denote the probability of matching of the primary numbers as , standing for syndicate probability. Probabilities of prizes The probability of matching primary numbers is given by Each of these outcomes the number of prizes won is known. For example, when and two of the primary numbers are matched with 6 of the winning numbers, then there will necessarily be 4 tickets matching three of the winning numbers. This is demonstrated in figure 1. Here, the winning numbers drawn are 1, 2, 15, 19, 25 and 40. Two primary numbers are correct, 1 and 2, whilst the other four primary numbers are not matched with any of the remaining four winning numbers. The remaining four winning numbers do match four of the secondary numbers, resulting in four 3-ball prize winning tickets. From the formula for we can calculate the probabilities of the various outcomes. For example, for a 6-ball win, all five of the primary numbers must be matched with only 1 of the 44 secondary numbers also correct, so the probability of winning the jackpot once is . This reduces to , where is the probability of a single ticket winning the jackpot as calculated above. This demonstrates that the probability of winning the jackpot using the e-lottery system is the same as for 44 random, distinct tickets. To get 5 balls correct, one can do it two ways: (a) match 5 primary numbers right with probability . Here, you will actually win one jackpot prize, one 5+ prize and 42 5-ball prizes. (b) match 4 primary numbers, the fifth not matching the bonus ball, with probability . Here you will match two 5-ball prizes. Thus, the total probability associated with winning 44 5-ball prizes is , since if the group has two tickets matching 5 balls and if the group has 42 tickets matching 5 balls. These sum to , which is just 44 times the probability of getting 5 balls right with a single ticket. Again this demonstrates that the probability of winning a 5-ball prize if one buys 44 syndicated tickets is the same as it is if one buys 44 random distinct tickets.

"Match 5 primary numbers right with probability. Here, you will actually win one jackpot prize ...."

Expected number of prizes For an event with a probability of 0.820, the odds of that event (given by ) give an indication of how often that event will occur. Thus, a lottery player using the e-lottery system will expect to win a prize every 1.22 draws he or she is entered in to. That player buying a single ticket will have to wait for around 54 draws on average to win a prize. Note that the increased prize frequency experienced for the e-lottery system would also be experienced if the player bought 44 random distinct tickets Systems for buying lottery tickets in syndicates cannot change the expected value of a ticket. However, syndicate systems do alter the variance in return. Of the alternative systems discussed here and with respect to the single player, a syndicate buying 44 random distinct tickets will have the lowest variance; the e-lottery system will have a higher variance; a single ticket will have the highest variance. It is possible to increase the expected value of a ticket by selecting unpopular combinations of numbers. Unfortunately, it is not currently possible, given the data available, to identify unpopular combinations. However, it is possible to identify unpopular numbers. It is known that lottery players have certain biases towards favoured numbers. For example, it is thought that players like to choose birthdays. As there are only 12 months in a year and 31 days (maximum) in a month, numbers above 31 are chosen less frequently than numbers below 31. One effect of this is that when high numbers are drawn as the winning numbers there may be fewer winners of prizes. Each of the players who do match the high numbers will share their winnings between fewer other winners, resulting in higher returns. Identifying unpopular numbers is addressed in Baker and McHale (2009), “Modelling the probability distribution of prize winnings in the UK National Lottery: consequences of conscious selection”, Journal of the Royal Statistical Society, Series A. For example, the 7 numbers picked least often are 46, 37, 35, 41, 49, 45. Choosing these numbers may increase the expected value of a ticket. Of course, the probability of winning a prize remains the same. Summary In this report we have looked at the probabilistic properties of the e-lottery system for the UK National Lottery game, Lotto. We have compared buying a single ticket to buying 44 tickets using the e-lottery system. There are two key results: 1. In terms of probability of winning any of the prizes, using the e-lottery system of 44 tickets is exactly the same as buying 44 random distinct tickets; 2. The expected number of prizes won when playing with the e-lottery is 44 times that of a single ticket.

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