Class is called Math for Liberal Arts, a college course. Here is the
question on my practice quiz:

A lottery drawing consists of choosing 6 numbers between 1 and 30 (any
order). If you choose 6 numbers between 1 and 30, what is the
probability that EXACTLY 4 of the numbers you choose are correct?

The answer is 0.00697

I do not know how they derived this answer.  I have tried many
different ways and can not come up with the answer they got.  I tried
this: 30 C 4 x 26 C 2 divided by 30 C 6.  This way doesn't seem to
work, so I must not be on the right track.  I have similar problems
like this as well.  Can you help?  I need it right away today, sorry
for the rush.

Hi Arturo,

Thanks for your question.  I will assume that you are familiar with
the combinatorial function since you make use of it in your question.
I will follow you in denoting the function with 'C'.  Just as a
refresher, the combinatorial function works as follows:

y C x = y! / (x! * (y-x)!)

It's also useful to note that Excel has a combinatorial function.  To
get 30 C 6 you would type '=combin(30,6)' into a cell.

OK, enough of the preliminaries...

**Important**: For any lottery type problem in which they ask the
probability of matching exactly x numbers after choosing y from a pool
of n, the formula will be:

(y C x * n-y C y-x) / n C y

So in the example you provide, we want the probability for matching
exactly 4 numbers after choosing 6 from a pool of 30.  The resulting
formula is:

6 C 4 * 24 C 2 / 30 C 6

6 C 4 is equal to 15.  This means that there are 15 ways of including
4 of the 6 correct numbers.

24 C 2 is equal to 276.  This means that there are 276 ways of
including 2 of the 24 incorrect numbers.

30 C 6 is equal to 593,775. This means that there are 593,775 ways of
arranging the 30 numbers into groups of 6.

Finally, (15 * 276) / 593,775 = .00697.  So given that there are 15
ways of arranging 4 correct numbers in a draw of 6 and there are 276
ways of arranging two of the incorrect 24, we multiply 15 by 276. 
This gives us all the possible ways in which we can draw 4 correct
numbers and 2 incorrect numbers.  We then divide that number by the
total number of possible ways of drawing 6 from 30: 593,775.

The probability of exactly 4 of the 6 numbers drawn from a pool of 30
being correct is .00697.

You might find the following websites helpful:

Info on the combinatorial function:
http://www.wizardofodds.com/games/pokerodd.html

Info on calculating lottery probabilities:
http://www.math.mcmaster.ca/fred/Lotto/

More examples:
http://www.wizardofodds.com/games/lottery-probability.html

If anything is unclear, please request clarification prior to
submitting a rating.

Best of luck!

-Blinkwilliams-ga

Another outstanding job, thank you.