Many players often ask how many Lotto numbers should be marked (i.e., which Lotto system to choose) to, for example, achieve a 3-hit.
There are different angles to this. On our installation page for the MELM program we presented
the 163-line guaranteed system that guarantees at least one 3-hit in every draw.
But you can also ask:
What is the minimum number of numbers one should choose on average (for a Lotto system) so that the probability of having, say, three correct numbers among those chosen is maximized? Mathematicians describe this using an expected value:
“How many balls must be drawn on average from 49 to obtain the third correct among the already known 6 winners?”
We are not talking about guaranteed systems here, but rather how many system numbers a Lotto system should include so that over hundreds or thousands of draws it will, on average, ‘catch’ 3 correct numbers (here the term “net” is actually fitting).
Here is the answer:
You should choose at least 22 system numbers.
More precisely, the expected value EX is 21.42857 numbers.
By coincidence, the German Lotto-Toto block offers LOTTO partial system VEW 622 (22 numbers in 77 lines). It also has guarantee level 3 with 3 correct and thus “fits” this purpose. NEW: On that page you can now “try” this, called Lotto partial system 22/77, with random numbers and recent official payouts (naturally without stake or winnings).
To clarify: by the law of large numbers, over several hundreds of draws a system with 22 numbers will on average yield a 3-hit. In a single draw it can happen that none of the 6 current winners is among your 22 numbers.
You can compute expected values for other target counts similarly:
EX ≈ 7.14 for 1 correct number,
EX ≈ 14.29 for 2 correct numbers,
EX ≈ 21.43 for 3 correct numbers,
EX ≈ 28.57 for 4 correct numbers,
EX ≈ 35.71 for 5 correct numbers,
EX ≈ 42.86 for 6 correct numbers.
These values are interesting; we could not quickly find them online, so it’s not impossible we made a mistake.
A quick check with MATLAB yields similar results:
% Simulates Lotto draws and
% computes average draw length for X hits.
X=3;
% number of simulated draws
n=500000;
% 6 “hits”, 43 “misses”
tip=[ones(1,6),zeros(1,43)];
av=0;
for k=1:n
% random permutation
permtip=tip(randperm(49));
cumtip=cumsum(permtip);
% first index reaching the X-th hit
len=find(cumtip>=X,1,'first');
% accumulate expected value
av=av+len/n;
end
% output
fprintf('Average length: %f\n',av);
