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Delanceyplace: Things You Didn't Know You Didn't Know

In today's selection -- though the average person is more likely to drop dead within one hour of purchasing a lottery ticket than to win the lottery, here are two strategies for you to consider, suggests John D Barrow.

The UK National Lottery has a simple structure. You pay £1 to select six different
numbers from the list 1, 2, 3, . . . , 48, 49. You win a prize if at least three
of the numbers on your ticket match those on six different balls selected by a
machine that is designed to make random choices from the 49 numbered balls. Once
drawn, the balls are not returned to the machine. The more numbers you match, the
bigger the prize you win. Match all six and you will share the jackpot with any
others who also share the same six matching numbers. In addition to the six drawn
balls, an extra one is drawn and called the 'Bonus Ball'. This affects only those
players who have matched five of the six numbers already drawn. If they also match
the Bonus Ball then they get a larger prize than those who matched only the other
five numbers.

What are your chances of picking six numbers from the 49 possibilities correctly,
assuming that the machine picks winning numbers at random? The drawing of each ball
is an independent event that has no effect on the next drawing, aside from reducing
the number of balls to be chosen from. The chance of getting the first of the 6
winning numbers from the 49 is therefore just the fraction 6/49. The chance of picking
the next of the remaining 5 from the 48 balls that remain is 5/48. The chance of
picking the next of the remaining 4 from the 47 balls that remain is 4/47. And
so on, the remaining three probabilities being 3/46, 2/45 and 1 /44. So the probability
that you pick them all independently and share the jackpot is:
6/49 X 5/48 x 4/47 x 3/46 x 2/45 X 1/44 = 720/10068347520

If you divide this out you get the odds as 1 in 13,983,816 -- that's about one
chance in 13.9 million. If you want to match 5 numbers plus the Bonus Ball, then
the odds are 6 times smaller, and your chance of sharing the prize is 1 in 13,983,816/6
or 1 in 2,330,636.

Let's take the collection of all the possible draws -- all 13,983,816 of them --
and ask how many of them will result in 5, or 4, or 3, or 2, or 1, or zero numbers
being chosen correctly. There are just 258 of them that get 5 numbers correct, but
6 of them win the Bonus Ball prize, so that leaves 252; 13,545 of them get 4 balls
correct, 246,820 of them that get 3 balls correct, 1,851,150 of them that get 2
balls correct, 5,775,588 of them get just 1 ball correct, and 6,096,454 of them
get none of them correct. So to get the odds for you to get, say, 5 numbers correct
you just divide the number of ways it can happen by the total number of possible
combinations, i.e. 252/13,983,816, which means odds of 1 in 55,491 if you buy one
lottery ticket. For matching 4 balls the odds are 1 in 1,032; for matching 3 balls
they are 1 in 57. The number of the 13,983,816 outcomes that win a prize is 1 +
258 + 13,545 + 246,820 = 260,624 and so the odds of winning any prize when you buy
a single ticket are 1 in 13,983,816/260,624, that is about 1 in 54. Buy a ticket
a week with an extra one on your birthday and at Christmas and you have an evens
chance of winning something.

This arithmetic is not very encouraging. Statistician John Haigh points out that
the average person is more likely to drop dead within one hour of purchasing a ticket
than to win the jackpot. Although it is true that if you don't buy a ticket you
will certainly not win, what if you buy lots of tickets?

The only way to be sure of winning a lottery is to buy all the tickets. There have
been several attempts to use such a strategy in different lotteries around the world.
If no jackpot is won in the draw, then usually the unwon prize is rolled over to
the following week to create a super-jackpot. In such situations it might be attractive
to try to buy almost all the tickets. This is quite legal! The Virginia State Lottery
in the USA is like the UK Lottery except the six winning numbers are chosen from
only 44 balls, so there are 7,059,052 possible outcomes. When the jackpot had rolled
over to $27 million, Australian gambler Peter Mandral set in operation a well-oiled
ticket buying and printing operation that managed to buy 90% of the tickets (a failure
by some of his team was responsible for the worrying gap of 10%). He won the rollover
jackpot and went home with a healthy profit on his $10 million outlay on tickets
and payments to his ticket-buying 'workers'.

Author: John D. Barrow
Title: 100 Essential Things You Didn't Know You Didn't Know
Publisher: W.W. Norton & Company
Date: Copyright 2008 by John D. Barrow
Pages: 152-154

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