Basic probability theory

The majority of the questions I receive deal more with basic probability theory than the material on the site. As the entire site depends heavily on the mathematics of gambling I've decided that a basic coverage of probability theory and an explanation of the terms used on the prediction pages might be helpful to many. I strongly believe that a understanding of this material is neccessary not only for proline betting but for any form of gambling.

The attempt to cover probability theory below is extremely basic but should be all you need to get started. If you want to further research on this subject, a web search will lead you to many sources which are much more qualified for this than I am, and cover the matter in greater detail.


I'm sure everyone understands what a probability is but I'll do a short explanation regardless. When we refer to the probability of an event we mean the chance of that event occurring out of a total possible number of outcomes. For example the probability of a coin flip being heads is 1 in 2. Two being the number of total outcomes and 1 being the number of outcomes we are discussing. This is also often expressed as a fraction of 1 where 1 represents the total possible outcomes. In this case the probability of heads would be 0.5. Or as a percentage of total possible outcomes - the probability of heads would be 50%.

If instead of a coin we were looking at the chance of rolling a specific number on one die the total number of possible outcomes is 6. As all the possiblities are equal the probability of a specific number would be 1 in 6, or .1667, or 16.67%.

I've used simple examples where the probabilities are the same for each possible outcome and therefore easy to express in terms of their relative probabilities. In proline wagering, or sports betting in general, this is of course not the case. It is still essential to find a method which assigns probabilities to the various possible outcomes. Simply saying that this side should win because _____ is just not good enough. In order to make a decision on which side is the correct wager you need to be able to set a probability of the event occurring. The reasons for this will be apparent in the section dealing with win expectancy.

Correct Odds

The correct odds, or sometimes referred to as the fair odds, are the odds at which a bettor would expect to break even over the long run. To continue our example of a coin flip the correct odds would obviously be 1 to 1. If you bet heads and lost $1 every time you were wrong while winning $1 every time you were right, since the probability of winings is 1 in 2, over the long run you would break exactly even. If calling the exact number on a die, the correct odds would be 5 to 1. Out of every 6 throws you would expect to lose 5 times for a net loss of $5 while winning $5 the one time you were right.

Again extremely simple in these cases. For more complicated probabilities the following formula can be used to give the correct odds.

WinProbability = Wins / TotalOutcomes (Wins is the favourable outcomes and TotalOutcomes is the sum of all possible outcomes)
CorrectOdds = (1 - WinProbability) / WinProbaiblity

Or using the numbers from our die example.
WinProbability = 1 / 6 = 0.1667
CorrectOdds = (1 - 0.1667) / 0.1667 = 5 (to 1)

Throughout this page when we refer to odds we are using the more common approach of basing the payout as a win TO risk ratio. It is important to be aware that proline odds are expressed as a return of win and wager. In other words a payout of 6.00 on proline is really 5 to 1 odds. When calculating the Correct Odds for proline purposes simply add 1 to the Correct Odds derived with the above formula. For example the correct payout (in proline terms) of our die game would be 6 FOR 1 or 6.00.

Expected Value

It is the combination of the probabilities and odds of an event that provide the expected value (usually referred to as the Player Edge on the site) of a wager. I cannot stress too much how important this number is. The correct betting strategy depends on the expected value, not on which result is the most probable winner!

Let's return to the coin flip to use as an example for our explanation of expected value. The probability of heads occurring we know is 0.5. Let's say I allow you to bet on heads, every time you lose you pay $1, but every time you win I pay you $1.10. as you will win half the flips you'll see a nice profit over the long run. The expected value is the profit made as a percentage of the total wagered. In this case after 100 flips you would lose $50 on the tails results but win $55 on the heads results for a profit of $5. This is the expected value - $5 profit on $100 net wagers - or +5%.

As another example lets play a game where your probability of winning was only 37% but I offered to pay you 2 to 1 on each win. After 100 trys you would expect to lose 63 times at $1 each and win 37 times at $2 each for a net profit of $74 (37 * 2) - $63 = $11. Your expected value in this case would be +11%.

For more complicated situations the expected value can be calculated using the following formula.

WinProbability = Wins / TotalOutcomes
LossProbability = 1 - WinProbability
ExpectedValue = (WinProbability * OddsToOne) - LossProbability * 100

Or using the numbers from our 37% example.
WinProbability = 37 / 100 = 0.37
LossProbability = 1 - 0.37 = 0.63
ExpectedValue = (0.37 * 2) - 0.63 * 100 = 11%

In the above example we have a case where although you were betting on an outcome that was a probable loser the expected value tells you that this is the correct wager. This is important to understand as it is a situation which occurs all the time in proline betting. It is not important which side is the probable winner, only what the value of the wager is!

Kelly Criterion

The Kelly method is at its simplest a betting strategy. However unlike the mulitude of betting strategies available, Kelly does not attempt to turn a sequence of negative expectancy wagers into an overall postive outcome. Kelly determines the mathematical optimal bet size of a wager which has a positive expectancy.

To use Kelly wagering you should first establish a bankroll which is completely set aside from your 'normal' money and is exclusively used for your proline wagering. Any winnings are added to that bankroll so that it can grow as rapidly as possible. The Kelly bet is the amount that increases the bankroll size at the optimal rate.

The Kelly rate is always a percentage of the net bankroll. The rate for any individual bet is dependent on both the size of your advantage and the probability of winning.

There are several formulas that can be used to approximate the Kelly Rate and I'll just give one here. To use this formula the expected value (ev) (as a %) should first be calculated as described in the previous section. OddsToOne represents the odds (on a to 1 ratio) of the potential payout. Just to repeat - the expected value must be positive or the result returned by this formula is meaningless.

KellyRate = (WinExpectancy/100) / OddsToOne.

The correct bet size is then Bankroll * KellyRate

When using Kelly it is important not to overestimate your advantage. Underbetting Kelly will lead to a slightly slower growth of your bankroll. Overbetting Kelly can lead to disaster. When in doubt err on the side of caution. Many people, myself included, find that Kelly betting is a bit too much like being on a roller coaster ride. For the sake of my nerves I prefer to aim for a half Kelly bet size. It's true that my bankroll growth will be slower but I'll accept that for peace of mind.

Although Kelly works great for sports betting and other forms of gambling there are problems in applying it to proline wagering. Everything would be fine if we wanted to play one parlay every day. Then the correct wager size could be easily determined. In reality we all play numerous parlays every day, usually in the form of some rotation. The problem is that the same games will overlap in the different parlays. Since the parlays are not independent of each other, determining the Kelly rate for each parlay and then betting that amount would be incorrect. Still finding some variation of Kelly which works would be preferrable than not using the method at all. A simple approach for example would be to determine which of your parlays has the largest bet size. Use that number for your total 'to bet' and then scale down all the indivual parlay's Kelly bet sizes down porportionally to meet that total. This would leave you underbetting Kelly by quite a bit but as mentioned earlier that may not be such a bad thing. I'm working on a method which attempts to more accurately set the bet sizes for an array of parlays and will provide that here at a future time.