Calculation of probability giving weird values

Question

I have a lottery of 60 numbers to chose from, 6 winning numbers are selected, with no repetitions. You win if you get 6 (jackpot), 5, 4, 3, 2 or 1 of the correct numbers.

You can chose to play 6 numbers(minimum), 7, 8, 9, ... up to 15 numbers.

I'm trying to calculate the odds (1 in x) of getting 1, 2, 3 up to 6 numbers in all different modes of the game, 6, 7, 8, ... 15.

For the first ones it's easy:

C60,6 = odds of winning the jackpot with 6 numbers bet
C60,6 / ( C6,5 * C54,1) = odds of winning the 5s with 6 numbers
C60,6 / ( C6,4 * C54,2) = odds of winning the 4s with 6 numbers
C60,6 / ( C6,3 * C54,3) = odds of winning the 3s with 6 numbers
C60,6 / ( C6,2 * C54,4) = odds  of winning the 2s with 6 numbers
C60,6 / ( C6,1 * C54,5) = odds of winning the 1s with 6 numbers

Then for the game with 7 numbers is similar

C60,6 / ( C7,6) = odds  of winning the jackpot with 7 numbers
C60,6 / ( C7,5 * C53,1) = odds of winning the 5s with 7 numbers
C60,6 / ( C7,4 * C53,2) = odds of winning the 4s with 7 numbers
C60,6 / ( C7,3 * C53,3) = odds of winning the 3s with 7 numbers
C60,6 / ( C7,2 * C53,4) = odds of winning the 2s with 7 numbers
C60,6 / ( C7,1 * C53,5) = odds of winning the 1s with 7 numbers

And so on up to 15

C60,6 / ( C15,6) = odds of winning the jackpot with 15 numbers
C60,6 / ( C15,5 * C45,1) = odds of winning the 5s with 15 numbers
C60,6 / ( C15,4 * C45,2) = odds of winning the 4s with 15 numbers
C60,6 / ( C15,3 * C45,3) = odds of winning the 3s with 15 numbers
C60,6 / ( C15,2 * C45,4) = odds of winning the 2s with 15 numbers
C60,6 / ( C15,1 * C45,5) = odds of winning the 1s with 15 numbers

But the values I get don't make sense.

For instance the values I get for the 1s is: N is the number of numbers played.

N   Prob
6   2.638 [C60,6 / ( C6,1 * C54,5)]
7   2.492
8   2.408
9   2.368
10  2.363 [C60,6 / ( C10,1 * C50,5)]
11  2.387
12  2.436
13  2.511
14  2.609
15  2.732 [C60,6 / ( C15,1 * C45,5)]

This doesn't make sense. I have a greater chance of getting just 1 number right by playing 10 numbers then I have playing 11, 12, 13, 14 and 15. And 15 is the worst of them all, even 6 numbers is better.

What am I doing wrong?

Here's a visualization of the probabilities.

Answer 1

This makes perfect sense. You're not computing the odds of getting at least one number correct; you're computing the odds of getting exactly one number correct. Given that one tenth of the numbers $1$ through $60$ are selected by the lottery, you'd expect that your chances of getting exactly one number correct are maximized when you choose ten numbers. And that turns out to be true, as you found.

Consider what would happen if you chose $56$ numbers. What then would be your odds of getting exactly one number correct?

Answer 2

The reason that it's less likely to get exactly one number correct when you choose 15 numbers as opposed to when you choose 10 numbers is that you have increased the likelihood of getting multiple numbers correct to the extent that getting only one number isn't as likely.

For example, if you would be allowed to choose 56 numbers, it would be impossible to only get one number correct.

Answer 3

For $10$ choices out of $60$ I expect on average $1$ correct. For $15$ choices out of $60$ I expect on average $1.5$ correct.

So it is not much of a surprise that the probability of getting exactly $1$ correct from $10$ choices is higher than the probability of getting exactly $1$ correct from $15$ choices

I get approximately the following probabilities

  correct   0       1       2       3       4       5       6
choices                             
6       0.5159  0.3790  0.0948  0.0099  0.0004  0.00001 0.00000002
7       0.4586  0.4012  0.1228  0.0164  0.0010  0.00002 0.00000014
8       0.4067  0.4153  0.1514  0.0247  0.0019  0.00006 0.00000056
9       0.3597  0.4223  0.1797  0.0349  0.0032  0.00013 0.00000168
10      0.3174  0.4232  0.2070  0.0470  0.0051  0.00025 0.00000419
11      0.2793  0.4190  0.2328  0.0607  0.0078  0.00045 0.00000923
12      0.2451  0.4104  0.2565  0.0760  0.0112  0.00076 0.00001846
13      0.2145  0.3983  0.2779  0.0926  0.0154  0.00121 0.00003428
14      0.1871  0.3833  0.2966  0.1104  0.0207  0.00184 0.00005998
15      0.1627  0.3661  0.3125  0.1290  0.0270  0.00270 0.00009997

and their reciprocals

correct 0    1    2       3       4        5         6
choices                             
6       1.9 2.6 10.6    100.9   2332    154518  50063860
7       2.2 2.5  8.1     61.1   1038     44981   7151980
8       2.5 2.4  6.6     40.5    539     17192   1787995
9       2.8 2.4  5.6     28.6    312      7791    595998
10      3.2 2.4  4.8     21.3    195      3973    238399
11      3.6 2.4  4.3     16.5    129      2211    108363
12      4.1 2.4  3.9     13.2     90      1317     54182
13      4.7 2.5  3.6     10.8     65       828     29175
14      5.3 2.6  3.4      9.1     48       544     16671
15      6.1 2.7  3.2      7.8     37       370     10003