Here comes the funny part. Maths and Gambling. Really like it, definitely suck at it (the second one!). The eternal delima to gamble or not. Or to be more rational, is it worth it to gamble? Let's start with the basics. Most of the times NOT! You think casinos were build up by the restoraunt's income? No, their main income source is the money that players loose.
Imagine an extraterrestial casino with allien owners. Despite our believes they cant' even count up to ten. Let's call that place Weirdpedia.Inside the Weirdpedia, there is this dice table which pays you 2 times your bet if you hit at least once the "one", in four times single dice roll. Maybe the name Chavalier or even Pascal comes up now, if not, thats ok move on.
The question is, should we play the game or not? We don't really care about the fun part of it, we just believe in money and we want more of them!
Motivation is prerequisited for success!
Time for some calculations before we sit on the table. If we roll a single dice four times there are 1296 possible outcomes. That's because the very first time we roll the dice, six outcomes may occur : (1,2,3,4,5,6). We do that one more time and the possible outcomes are 36. They are not so many so let's write them down to be easier for us to understand.
$(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) $
$(2,1) (2,2) (2,3) (2,4) (2,5) (2,6) $
$(3,1) (3,2) (3,3) (3,4) (3,5) (3,6) $
$(4,1) (4,2) (4,3) (4,4) (4,5) (4,6) $
$(5,1) (5,2) (5,3) (5,4) (5,5) (5,6) $
$(6,1) (6,2) (6,3) (6,4) (6,5) (6,6) $
At this point you should be able to understand that four times dice roll will output 6x6x6x6=1296 possible combinations (multiply by 6 for every time you roll). Now we have to see what the odds are. In once dice roll game, we have a chance out of six to win. In fraction this is 1/6 , or about 16.7% ratio to win. In other words if we play six times, in one of them we expect to win.
Let's say we place 1 weirdollar each time we play, and do that six times, of course 6 wirdollars will leave out our pocket. For the single win we made, 2 wirdollars will come back. I think we are four wirdollars missing and i don't like that!
Ok, now I switch table and go to one that is twice dice rolling. Above we wrote down possible outcomes and they are 36. Out of them, good for us are 11 of them (indicated with bolds). So the possibility to win is 11/36 or 30.5%. That means if we play ten times (minus 10 weirdollrs) 3 we will win (3x2=6 weirdollars). So again, we are 4 weirdollars missing! Ohhhh you alliens, you just messing with us. Note that in single dice roll time the chance to win was 16.7%, but in twice dice roll we don't get 2x16.7%=33.4%. What happened? This is a conditional probability as we say in math. For now all we have to understand is the reason why that happens. As you may noticed at the previous matrix that we wrote the outcomes, we have the sets (2,1) and (2,1), or (5,4) and (4,5) etc, so these appear twice. But in the diagonial of the matrix we get only (1,1), (2,2), (3,3)... different game, different outcomes, different odds!
Finally its time to move to the four times dice roll table. Show me the money!
Ok, we already know that possible compinations are 1296, but how many of them are profitable? Because 1296 combinations are too many to write them all and after that just count the 'good' ones we will try something different. We can find how many of them don't win.
A combination that doesn't win, doesn't have 'one' inside it. So, we remain with five possible outcomes. 5 for the first time we roll by 5 the second time we roll by...this yields 5x5x5x5=625. Those combinations don't win.
1296 are total, so 1296-625= 671 win. Nice, they are many!
The odds to win are 671/1296 or in percentage about 51.8%
In other words this means every time we play, once we loose and once we win.
We play twice so -2 weirdollars
and we win once +2 weirdollars. Total income ZERO. Shame on you bloody alliens!
As we have seen, this game doesn't worth playing it. Even though odds are just little higher that 50-50, just like coin flip, i should inform you that there a legend about a doomed guy, who during his prison time was coin fliping alone and marked the outcomes. The distribution of outcomes normalized at about 10000 (yes, ten thousand) flips. In our case, this means that after 10000 plays we will have no loss, but also no winnings at all. Totally waste of time!
In case that you find a table that lets you roll the dice five times, will it worth it or not?
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