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### NCAA Lottery Formula

```
Date: 08/27/97 at 13:56:59
From: Anonymous
Subject: Lottery formula

Hi,

I was surfing the net to find the formula to a math question and
thought maybe you could help me. This is a probabilities/odds
question. I have forgotten the formula.

I recently won the NCAA final lottery enabling me to purchase two
tickets if I wish. I'd like to know the odds of my winning because I
have won both times I have entered.

There were 140,000 entries for 10,000 tickets. (I think you have to
assume that every entry requested two tickets (the max) thus making it
really 5000 winning entries.) I entered it ten times (the max one
could enter). I'd like to know the odds of prevailing on that.

Also, my secretary entered it ten times under her name on my behalf.
She didn't win. How did this change my chances? Presumably it doubled
them.

Finally, assuming the same 140,000 for 10,000 tickets two years ago,
for the final 4 in Seattle, what are the odds of my winning twice (I
only entered one entry that time and had no "secretarial entries").

My logic goes something like this with respect to winning this year
only.

5000/140000 = 1/28    (if I  had only entered once)

20/140000 = 1/7000  (if I had entered 20 times (with my
secretary's help) but only one winning entry
could be drawn)

Do I have a 1/7000 chance 5000 times in a row?

I know there is an easy formula for this. Can you help?

Thanks, Ed
```

```
Date: 09/02/97 at 09:21:21
From: Doctor Rob
Subject: Re: Lottery formula

You have started down the right path.  The chance of winning with one
entry out of 140000 is 5000/140000 = 1/28, or 3.5714 percent.

The chance of one or more winning entries having submitted 20 is
1 - (1 - 5000/140000)^20 = 51.6814 percent. (Think of the chance of
none of the entries winning, and subtract that from 1.)

The chance of one or more winning entries having submitted 10 is
1 - (1 - 5000/140000)^10 = 30.4884 percent.

The chance of exactly one winning entry having submitted 20 (both of
you) is 20*(5000/140000)^1*(1 - 5000/140000)^19 = 35.7916 percent.

The chance of exactly one winning entry having submitted 10 (you
alone) is 10*(5000/140000)^1*(1 - 5000/140000)^9 = 25.7450 percent.

The chance of no winning entries having submitted 10 (your secretary)
is (1 - 5000/140000)^10 = 69.5116 percent.

The combined chance of winning last year with one entry and winning at
least once this year with 20 entries is

5000/140000*[1 - (1 - 5000/140000)^20] = 1.8458 percent.

Consider yourself very lucky last year, but about what you might
expect this year.

By the way, if the probability is p, the odds of winning are
k*p to k*(1-p) for any real k. For example, if the probability is
1/28, then the odds in favor are 1 to 27 (k = 28), and the odds
against are 27 to 1.

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

```
Date: 09/03/97 at 09:28:46
From: Anonymous
Subject: Re: Lottery formula

Thanks very much for the answer!  Now I'm trying to figure it out.
What is the "^" sign?  Thanks.
```

```
Date: 09/03/97 at 10:45:36
From: Doctor Sarah
Subject: Re: Lottery formula

Hi -

In e-mail, the ^ sign means "raised to the power of"; thus 2^2 is
2 squared, 10^20 is 10 to the 20th power, etc.

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

```
Date: 09/05/97 at 13:45:34
From: Anonymous
Subject: Re: Lottery formula

Thanks again for the response.  I never knew that was what ^ stood
for.

Ed C.
```
Associated Topics:
High School Probability

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