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It's straightforward but takes a little math.
if you mark L spots, the probability of hitting exactly R of them is given by the formula:
C(L,R) * C(80-L,20-R)
p(L,R) = -------------------------------
C(80,20)
The expression C(X,Y) represents the number of possible ways to select Y items from a larger collection of X items, where order of selection is unimportant. Many calculators, spreadsheets and math libraries have a built-in facility for calculating this function. Both Lotus 1-2-3 ™ and Excel ™ name this funcion COMBIN(n,r); it is also known as the "binomial coefficient" function.
Let's use the formula to calculate our chances for hitting a 6-spot:
p(6,6) = C(6,6) * C(74,14) / C(80,20) ~= 0.00013 or 1 in 7753.
Rarely do we hit all 6. Let's calculate the whole 6-spot table from the above formula. Note these numbers are independent of the house payoff as they are merely the probability of an event happening, regardless of whether any money is wagered.
p(6,6) = C(6,6) * C(74,14) / C(80,20) ~= 0.00013 Catch 6
p(6,5) = C(6,5) * C(74,15) / C(80,20) ~= 0.00310 Catch 5
p(6,4) = C(6,4) * C(74,16) / C(80,20) ~= 0.02854 Catch 4
p(6,3) = C(6,3) * C(74,17) / C(80,20) ~= 0.12982 Catch 3
p(6,2) = C(6,2) * C(74,18) / C(80,20) ~= 0.30832 Catch 2
p(6,1) = C(6,1) * C(74,19) / C(80,20) ~= 0.36349 Catch 1
p(6,0) = C(6,0) * C(74,20) / C(80,20) ~= 0.16602 Catch 0
Total 0.99932
The total should always be 1, or very close due to rounding, since one of the above outcomes will happen.
To find the probability of one of several outcomes, you add the numbers for each entry. In the above 6-spot table the chances of catching "exactly 0 OR exactly 1" are 0.52951. Meaning more than half the time you'll catch at most one number on your 6-spot ticket. Similarly, roughly 1 in 6 tickets will be a winner according to the payoff table presented above. (Since Catch3+Catch4+Catch5+Catch6 probabilities total 0.16159 or 1/6.19).
Of course the chance of having a winning ticket is not as interesting as knowing what the expected return is. This is calculated by adding together the expected return for every possible outcome. Now for any possible outcome you can calculate the expected return for that outcome by multiplying the payoff for that outcome by the probability of that outcome. This leads directly to the following set of calculations, using the 6-spot payoff and odds already presented:
Catch chance payoff expected
($1 bet) return
6 0.00013 1500 0.195
5 0.00310 50 0.155
4 0.02854 8 0.228
3 0.12982 1 0.130
2 0.30832 0 0
1 0.36349 0 0
0 0.16602 0 0
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Total 0.708
For this payoff schedule you can expect to receive a return of 71 cents for every dollar bet.
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