Fwd(2): Re: lotto

Sanderson Smith (Sanderson_Smith@cate.org)
Sun, 01 Nov 1998 10:03:18 -0800

For those who might want probability examples, here's something that might be useful. I received the note from Rich Turner,
responded to him, and sent a copy to my AP Stat. students since we have recently done an analysis of California Lotto.
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Hi Stat. students... I just received this note over the AP Statistics ListServe. To my knowledge, I have never met Rich
Turner. The question he asks provides a wonderful opportunity to review some important ideas. Please take a few minutes to
read through this correspondence. This does represent a great review sheet.
Doc
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richturn@wave.co.nz writes:
>Hi, I have just been reading your article on statistics. I am trying to find out how many combinations & what the combinations
>are for the New Zealand Lottery.
>There are 40 numbers & 6 numbers are drawn. I need to know all combinations.
>Is there any chance you could let me know how to work it out because i havent got a clue?
> eg. numbers 1-40 6 of
>
>A. 05, 11, 17, 25, 27, 33
>B. 03, 09, 14, 26, 35, 40
>C.01, 10, 18, 22, 37, 39
>D. 02, 04, 12 13, 15, 31 There is no limit to the amount of lines.
>E. 01, 05, 20, 23, 30, 40
>F. 08,09, 15, 24, 28, 36
>G.11, 17, 18, 22, 25,31
>H. 02, 04, 12, 19, 26, 33
>J. 07, 11, 17, 28, 29,32
>K. 01, 21, 26, 35, 37, 38
>L. 04, 09, 18, 23, 27, 30
>etc,etc,etc,etc
>Thank you for your help. This is for a project i am working on at college.
>Rich Turner.

Hi Rich...

Thank you for your note. I'm forwarding a copy of this to my AP Statistics students since this is "in line" with material we
have studied.

OK, you have 40 numbers, and 6 are chosen at random.
The number of possible combinations is
40C6 ("40 choose 6"), which, on a TI-83 calculator is 3,838,380

You can also get this by the following reasoning. If you pick 40 numbers randomly, the number of possible arrangements is
40x39x38x37x36x35
But, this number contains every every number combination 6! = 6x5x4x3x2x1 times, so you need to divide the above product by 6!
(6 factorial). If you do, you will once again get
3,838,380.

I definitely wouldn't attempt to write down all possible 6-number combinations from New Zealand Lotto. That would take quite a
bit of time.

Here's a bit more information that may or may not be useful to you.

Match 6: Number of ways = (6C6)(34C0) = 1
Match 5: Number of ways = (6C5)(34C1) = 204
Match 4: Number of ways = (6C4)(34C2) = 8,415
Match 3: Number of ways = (6C3)(34C3) = 119,680
Match 2: Number of ways = (6C2)(34C4) = 695,640
Match 1: Number of ways = (6C1)(34C5) = 1,669,536
Match 0: Number of ways = (6C0)(34C6) = 1,344,904

If you add up these numbers, you once again get 3,838,380.

Thanks again, Rich, for providing me with an opportunity to demonstrate MATH POWER to my Advanced Placement Statistics
students.

"We figured the odds as best we could, and then we rolled the dice."
(U.S. President Jimmy Carter, June 10, 1976)

Sanderson