Expert: Chanda Walker - 3/29/2009

Question
QUESTION: There is a lottery where there are 5 numbers drawn out of 45. You have the
option of picking 7, 9, and 10 numbers out of which you only need to match the
5 drawn numbers. What are your chances of winning, when you choose 7, 9 or
10 numbers. (Numbers are drawn from 1-45). Thanks

ANSWER: The way I think about this is that there are 5 red balls and 40 blue balls in a bag.  The red balls represent the 5 winning lottery numbers.  You want to draw ALL 5 out of the bag.  You get to pick draw 7, 9 or 10 numbers.  

Choose 7:

The number of ways to pick 5 red and 2 blue:

5C5 * 40C2 =  1 * 780

The number of ways to pick 7 out of 45 is:

45C7 = 45379620

The ratio of these are: 1/58179 or ~0.000017
******************
Choose 9:
The number of ways to pick 5 red and 4 blue:
5C5 *40C4 = 91390

The number of ways to pick 9 out of 45 is:
886163135

The ratio:
2/19393
**********************
Choose 10:
5 red and 5 blue:
658008

10 out of 45:
3190187286

The ratio:
4/19393


If you don't understand the 40C4 notation, just let me know and I'll explain that fully.





---------- FOLLOW-UP ----------

QUESTION: Thanks for the reply.

Firstly, yes I do need you to explain to me the 40C4 notation and secondly I
don't understand your response, when you say 2/19393 is that 1/9,696.5 or
~0.000103 and the same for choosing 10 and needing only 5 of these to win?

Thanks

Answer
40C4 is said:  40 choose 4

This is standard notation for permutation and combinations.

nCk  (n choose k) has a numerical formula of:

n!/((n-k)!k!)


In case you don't know what the ! is . . . that is called a factorial and I'll give you an example:

5! is 5*4*3*2*1

You just multiply all the integers starting from the number given down to one.  Another example:

7! = 7*6*5*4*3*2*1


When I say 2/19393 I mean the fraction of 2 divided by 19393.  I didn't divide that out all the way since I assumed you had a calculator handy.

Let me know if it still doesn't make sense.  If you want more fundamental understanding of what is going on, just let me know.  I never know how deeply one wants me to explain things when they only type the question.  If you want me to explain the concept, I can and I'm happy to do so.