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### Lottery Probability

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Date: 01/23/98 at 21:08:56
From: marco and diane
Subject: Lottery probability

We've been having a heated discussion about the probabilty of picking
winning lottery numbers. Let's say that Monday's winning numbers were
1,2,3,4,5, and 6. Is there or is there not the same probability that
those winning numbers will come up again? I mean isn't there the same
chance that any random numbers or those exact same numbers will be
repeated 3 days later? (The numbers to choose from are 1-46.)  In
other words what is the probability that Thursday's numbers will be
the same as Monday's numbers? (Diane's question.)

I know that the chance of those numbers coming up on any given day is
1 in 10 million. Now if you play the same numbers as the previous
winning numbers, doesn't the probability change?  (Marco's question)
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Date: 01/23/98 at 22:31:23
From: Doctor Budrow
Subject: Re: Lottery probability

Dear Marco and Diane,

There are two distinct questions here so you are probably arguing
about different probabilities. Often the hardest thing in probability
is clarifying exactly what the question is.

The first question is: what is the probability of one set of six
numbers being drawn at random if you know the outcome of a previous
draw?

The other is: what is the probability of the same set of six numbers
being drawn twice in a row?

An equivalent question (but with a simpler example) to the first one
is: what is the probability of heads on the toss of a coin knowing
that the previous flip was also heads?

An equivalent question to the second one is: what is the probability
of flipping two heads in a row?

In the case of the coin, the chance of flipping heads knowing the
previous flip was heads is 1/2, if the coin is fair. It would also be
1/2 if the previous flip were tails. The chance of flipping two heads
in a row is 1/4, as is the chance of flipping a heads then a tails or
a tails then a heads or a tails then a tails.

The difference between the two questions is that in one case you are
stating that each flip of a fair coin is independent of the previous
flip, and in the other you are asking what the probability of a
specific pair of outcomes is.

In the case of the lottery, Marco is asking what the chance of winning
with the same numbers that were played the week before is. This is the
same as the chance of flipping a coin and getting heads when you know
the previous flip was also heads, which is 1/2. So, yes, he has one
chance in 10 million of winning with the same numbers. But if you
choose a set of six numbers, the chance of that set being drawn twice
in a row is 1 in 10 million squared, a very small chance indeed.

One more way to look at this question is to ask: if I examine all the
winning combinations drawn so far, what is the probability that at
least one combination has been repeated?

-Doctor Budrow,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

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Date: 01/24/98 at 00:00:54
From: Doctor Bill
Subject: Re: Lottery probability

Marco and Diane,

Suppose you flip a penny one time. What is the probability of getting
a head? It's 1/2, right? Suppose you flip the penny a second time.
What is the probability of getting a head? It's still 1/2, isn't it?
Does the outcome of the first flip influence the outcome of the second
flip? Suppose you flipped a penny 10 times, and each time it came up
heads. On the 11th flip, what is the probability of getting a head?
Does the coin remember that it has come up heads 10 times in a row?
The probability of getting a head on the 11th flip of the coin is
still 1/2.

The lottery is the same sort of thing. If 1,2,3,4,5,6 came up on
Monday, what is the probability that 1,2,3,4,5,6 will come up on
Thursday? Do you think that 11,12,13,14,15,16 is more likely on
Thursday than 1,2,3,4,5,6?  Do the little lottery balls remember which
ones came up on Monday, and thus will let some of the others come up
on Thursday?

I think both of your questions are the same question. The probability
of Monday's numbers coming up again on Thursday is the same as for any
other numbers coming up on Thursday.

I think your confusion (and it is shared by many) might lie in the
fact that you are thinking that YOU pick some numbers and then hoping
THOSE numbers will come up twice in a row. (That is very unlikely.)
Remember, the first group of numbers is always picked for you in
Monday's lottery, and you have to match them on Thursday (or they are
picked for you on Thursday and you have to match them on Monday). In
either case, you are given numbers that you must match, so what is the
probability that they will come up in the next drawing? It's still 1
in 10 million!

-Doctor Bill,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability

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