Let’s say that this is a kind of story of a fisherman playing toto/lotto (depending on how it’s called at your area :) ). The general case of this game is a main set of “N” numbers out of which “n” are chosen fairly..., without repetition… However, since our fisherman is not very good in probabilities let’s try to give an example with numbers from the Bulgarian surroundings where there is such a game called 5 of 35 OR where our general set N=35 and the sample draw n=5 and for the protocol... in one game there are 2 independent 5/35 draws…
So trying to keep the things simple let’s try to present what happens within one such draw. Well using the long ago invented Arabic numbers (1,2,3…) and the rules of combining numbers or equal separable entities :), using Google sheets (or other tools…), we have combin(35;5)=324 632 possible combinations. Then using a little bit more complicated calculations for combining numbers and/or groups of them, we have the following distribution of parity:
N |
distribution |
g (=nr. elements) |
r (=nr. elem. perm.) |
comb. |
((N-n)+1)! |
fact g |
(31-g)! |
Probab % |
1 |
1+1+1+1+1 |
5 |
1 |
169911 |
8,2228E+33 |
120 |
4,0329E+26 |
52,34 |
2 |
2+1+1+1 |
4 |
4 |
125860 |
8,2228E+33 |
24 |
1,0889E+28 |
38,77 |
3 |
2+2+1 |
3 |
3 |
13485 |
8,2228E+33 |
6 |
3,0489E+29 |
4,15 |
4 |
3+1+1 |
3 |
3 |
13485 |
8,2228E+33 |
6 |
3,0489E+29 |
4,15 |
5 |
3+2 |
2 |
2 |
930 |
8,2228E+33 |
2 |
8,8418E+30 |
0,29 |
6 |
4+1 |
2 |
2 |
930 |
8,2228E+33 |
2 |
8,8418E+30 |
0,29 |
7 |
5 |
1 |
1 |
31 |
8,2228E+33 |
1 |
2,6525E+32 |
0,01 |
Sum |
324632 |
100,00 |
Where out of the all 324 632 possible combinations we have exactly 13 485 combinations of type (2+2+1 - ex 3,4,12,13, 33), exactly 13 485 combinations of the type (3+2 - ex 17,18,19, 26,27), exactly 169 911 combinations where we have no any parity (ex 5,7,19,23,34) and so for the other types of the parity distribution with its corresponding combinations.
Since the comprehension level of our fisherman is better with fish and other things than numbers, let’s try to help him understand a little bit better what the above table logically means. I would argue that in an incremental increase (defined as each next independent, fair...draw), each one of the 324 632 combinations should have an equal chance to occur… Imagine that each combination of 5 numbers corresponds to a caught fish consisting of 5 numbers put in a basket. So after catching the whole...lake :) we have a basket-full of equally shaped and sized 324 632 five-number-fishes...which are separable only by their content (I mean that every combination of 5 numbers is a unique 5 out of 35 set)... Then If we re-catch one of these “5-number fishes” from the basket using a random table help..., all of them should have an equal chance to be pulled out of the basket… Yet following the above calculated distribution of parity and applying the law of the big numbers, after having a big enough number of re-caughts from the basket, we should start having a distribution/diagram shaping in accordance with the above calculated table…I mean the diagram shall have its peak along the unpaired set of combinations, followed by those with one pair in them...etc..And this would be because most of the equally re-caught basket-full fishes (52,34%) consist of unpaired numbers, (38,77%) consist of one pair and three unpaired numbers and so on...
But as many can...feel?, our fisherman remains dissatisfied, because he still says - so what how would this knowledge help me play toto/lotto or catch fish better, so let’s see are there ways of improving the play/catch? Yes there are, for instance by improving the catching tools from a single hook (playing with 1 combination) to fishing simultaneously by manu hooks or a net :)
Hence as I believe many of you can guess that one’s chances to catch fish improve substantially if he/she fishes with let’s say 6 meters radiused (round) net (the shape matters, but is not that important…) instead by a single hook… Well that could be compared to toto’s/lotto’s full or partial combinations games, which are often readily available… I mean that out of the 35 general-set numbers, one could for instance...choose 12, which combine into 5 in exactly 792 ways.
In our example that would mean that if we play with all such possible (comb(12;5)=792) combinations, our let’s say 12 meters diemetered net (the comparisons aim to close the gap between...combinatorics/probabilities and the fisherman’s level of understanding…), we shall be able to catch every single fish falling under its area…However that’s usually very expensive, so our fisherman decides to try with a less expensive 12 meters diameter net, which has bigger holes in its structure… So that would practically mean that some of the fishes falling under the area of the net would be able to escape, because of the...bigger holes in its structure...And that corresponds to playing toto/lotto by not-full range of combinations... I mean that in the given example of 792 combinations, we could chose less...for instance 30 :) and pay less for the net/game…(instead of 792xY we pay 30xY, where Y is the price of a single combination…).
Yet if one asks, could the fisherman increase further his chances for catch, I would generally say YES - by playing/fishing with different sizes and configurations of nets (the size and... the place of the holes might matter)... For instance if he/she inreseas the area of the...round (not compulsory…) net to ... 9 meters in radius... (A=πrr) :), he/she increases the chance of catching fish under the net/game…, as respectively the rule is “the more the most”...I mean the the more combinations you play with (a bigger net…), the more your chances for match :) increase… Here I intentionally used the word match not win, because the toto/lotto games are usually preliminarily tested and calculated so, that…for instance the Jackpot amount :) would be considerably less than the price paid for the full range of combinations in the game…
For instance according to the current Eurojackpot rules :) the maximum Jackpot could reach 90 mln euro and if someone wants to assure 7 numbers correct (not necessarily a full Jackpot, because there could be other players sharing it…), he must pay 2 x 95 344 200 = 190 688 400 euro, which is a match but not a win when the cost for playing is the full range of combinations...or such that is bigger than the prize-sum...
P.S. The story of the fisherman has a...kind of … sad real life background, which for now I shall just call “a tale that wasn’t right…”....
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