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### Lottery Strategy and Odds of Winning

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Date: 04/27/2000 at 16:25:58
From: Martin Smith
Subject: Odds of winning vs. Odds of Losing

My friend maintains that my odds of winning in the UK National Lottery
are the same whether I choose the same numbers each time or whether I
choose different numbers. I maintain that my odds of losing are
greater if I change my numbers than if I keep the same ones. The
theory is:

FACT: The odds of winning are approximately 13 million to one. So
according to my friend:

1. No set of numbers is any more likely to win than any other.
2. The balls and machines have no memory, therefore each draw is a new
one.
3. If you entered 13 million times you would win.

BUT I maintain:

1. If you change your numbers you may have a winning set the week
before that set wins. In other words, you could enter 13 million
draws and get the next week's winning numbers every time and still
lose every time.

2: If you enter 13 million times with the same numbers, you will win
because your number is bound to turn up for one of them.

I know this is not exactly the case as it is perfectly possible to
have the same numbers more than once; however, I believe this will
affect both scenarios in the same way.

So who is correct?
```

```
Date: 04/28/2000 at 15:59:30
From: Doctor TWE
Subject: Re: Odds of winning Vs Odds of Losing

Hi Martin - thanks for writing to Dr. Math.

Assuming that we are talking about playing the numbers on different
weeks (as opposed to buying multiple tickets for the same week), and
assuming that the generating machine is fair, then your friend is
right. No set of numbers is any more likely to win than any other.
This is correct if the device is fair. In probability, we usually
assume a fair random number generator, but in the "real world" this
may not be the case. Sometimes machines are deliberately rigged;
sometimes they just aren't perfectly designed.

2. The balls and machines have no memory, therefore each draw is a
new one.

This is absolutely correct. Each week's drawing is an independent
event, not affected by past or future drawings.

3. If you entered 13 million times you would win.

Not necessarily. On the _average_ you should win once in 13 million
drawings, but they may repeat numbers that aren't yours and not pick
yours. If it's a fair machine, in 13 million million drawings, you
should win about 1 million times.

This is like saying that if you roll a die 6 times, you'd get a 6
exactly once. This is not the case. On the average, 1/6 of the rolls
will be 1's, 1/6 of the rolls will be 2's, etc., but we could get two
4's and no 6. Likewise, we could get three 6's but no 2's. In enough
rolls, it will average out. Likewise, in enough lotteries (billions of
them), it will average out.

>1. If you change your numbers you may have a winning set the week
before that set wins. In other words, you could enter 13 million
draws and get the next weeks winning numbers every time and still
lose every time.

Say your number is 1-2-3-4-5. It is equally likely that every week,
the winning combination will be 6-7-8-9-10. Each week you will miss it
by "that much".

>2. If you enter 13 million times with the same numbers, you will win
because your number is bound to turn up for one of them.

See what I said about your friend's point 3. Not every number will
come up in those 13 million drawings.

An important concept that you're missing is the idea of replacement.
If, once drawn, the numbers are removed from the pool and cannot be
picked ever again, then sooner or later you would win in 13 million
drawings. If you stuck with the same number, sometime in those 13
million drawings your number would come up. If you changed numbers
each week (we'll assume you only choose from those that haven't yet
been picked), sooner or later you'll pick the winning one. If, by the
13 millionth drawing, you hadn't picked the winner yet, you'd know
which one was left.

This would be like guessing which card will be picked from a deck and
placed on a table. Your chances are 1 in 52 when the first card is
picked. Say that it's the 5 of clubs. If you're wrong, the card is
placed on the table (NOT reshuffled into the deck) and you try to
guess the next card. If you haven't guessed the card in the first 51
draws, you can figure out which card is left and guarantee that you'll
"guess" right on the 52nd draw.

But the lottery is not like that. Each time a combination is picked,
it is "reshuffled" into the mix and can be picked again in any of the
following weeks. The reason that we don't seem to "see" this is
because of the small sample size relative to the pool of possible
combinations. If we COULD play millions of millions of lotteries (we
can simulate this with computers), we'd see repetitions, and we'd see
that the odds of winning are the same for either strategy.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Probability

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