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@Thehammerwields is a mathematician who specializes in games of chance. A die-hard fan of Husker Football, New York Yankees Baseball, UNC Basketball, and Professional 501 Darts!

Greg Heim

Let’s say you wager $1.00 on a coin flip. There are only two possible outcomes:  heads and tails.  One will result in a loss, and the other will result in a win.  Each outcome has a success probability of 50%, so the total adds up to 100% or one – which it must in each and every case in order for the calculation to be correct.  The analysis of all of the possible results is referred to as an event cycle.  In our basic example, the total number of results is two. 

If I lose, my $1.00 is lost in its entirety.  If I win, I win $0.95 and get to keep the additional $1.00 that I wagered.   Because the amount of money won is less than the amount of money lost on all the possible outcomes, you don’t need mathematical analysis to see that this particular wager puts the player at a financial disadvantage. This financial disadvantage is referred to as negative expectation from the perspective of the person who is making the wager.

But negative expectation has much more than a qualitative (non-numerical) component; it cannot always be eyeballed.  Being able to calculate and do a quantitative (numerical) analysis to determine HOW MUCH negative expectation a wager has is of extreme importance.

To calculate the expectation for our coin flip game, we must wager the same amount of money for each flip, or trial and make that dollar amount our starting number – or in essence our initial bankroll.  In this case, our initial bankroll is $2.00 – a paltry sum but perfect to illustrate the concept.

If we lose the first flip, we lose $1.00 – so $2.00 - $1.00 is $1.00.  If we win the second flip, we win $0.95 cents – so $1.00 + $0.95 = $1.95.  

Because we started with $2.00, and only ended up with $1.95 – we are going to have an expected loss of $0.05 for every $2.00 wagered.  When we divide $0.05 by $2.00, the result (quotient) is 0.025, or 2.5%.  Hence, the player has a negative expectation of 2.5%.

It is important to note that the player can never overcome negative expectation in an honest game.  They cannot engage in such practices as a) discontinuing play after they win or lose a certain amount of money, b)  changing which result they pick, in this case heads or tails, c)  wagering constant or non-constant amounts of money on any trial, or d)  starting with ANY finite amount of money – just to name a few.  

The value of negative expectation is expressed in mathematical terms as a boundary inequality.  The negative expectation must be greater than zero and less than or equal to one (or 100%), The mathematical nomenclature is expressed as such:  

0 < x <= 1 (or 100%) – where x is the amount of negative expectation.


Gamblers can be placed into four categories:  1) They know the reality of casino gambling and care about making wagers which will lower their negative expectation as much as possible, 2) They know the reality of casino gaming, but for the most part make the wagers they want as they are willing to “pay the price,” 3) They don’t know, but they want to know, and 4) They don’t know, and they don’t want to know.

I came across a perfect example of our second example at the South Plainfield Community Pool several years ago.  I was approached by one of the patrons with a roulette question.  The individual mentioned which wagers they liked to make, and they knew there were other bets on the table which would cut the house’s advantage in half – but they wanted to be “entertained” and those other wagers were boring.  

This person also made numerous other comments which displayed a better-than-average understanding such as playing responsibility by setting a loss limit for their trip, and making sure that the money that they were using was purely disposable income.

But it was the following statement that floored me because it was such an anomaly:  “I know the casino is going to win $1.00 for every $19.00 they wagered on average (a negative expectation for the player of 5.26%), but if eat a jelly donut and pay $1.00 for it – your $1.00 is gone and that’s that.  When I play roulette, I could have a variety of short-term results that where I hit my loss limit, have a short but fun run of luck, or anything in between.”

Responsible and knowledgeable gambling has a very basic premise:  Knowing the REALITY of what you are facing while making choices of your own freewill.   However, it is that REALITY portion where the problem lies – because ignorance and misinformation ARE NOT bliss when it comes to having more of your hard-earned money leaving your wallet than it should while you are whining as to why “that” always happens!