What Are the Odds?

understanding_oddsSkill vs Chance


Lottery, Powerball, Big Game, Scratch tickets, Keno, Bingo, Roulette, Craps, Slots, Video poker, Baccarat

Skill (at least partial):

Blackjack, Poker, Handicapping horses, Sports betting, Pool, Bowling, Golf, Chess, etc.


(Reprinted with permission from Facing the Odds: The Mathematics of Gambling and Other Risks. Harvard Medical School, Division on Addictions.


  • Each possible outcome is that process has the same chance or probability of occurring
  • Outcomes are determined by chance

For example:
When flipping a coin, the probability of the coin landing heads up is the same as the probability of the coin landing tails up (50%). Similarly, when rolling a die, the probability of rolling a one is the same as the probability of rolling a two, which is the same as the probability of rolling any of the possible numbers.

The reason that all possible outcomes of events like these are equally likely is that the outcomes are determined by chance; there is no reason to believe that one result should occur more often than another result. Since the factors that determine the outcome on a single roll of a die are unknown and unpredictable, we can say that the outcome on any one roll is random.


  • The likelihood that a given event will occur
  • A number expressing the likelihood that a specific event will occur, expressed as the ratio of the number of actual occurrences to the number of possible occurrences (Soukhanov, 1992)

For example:
Suppose you were rolling a die and wanted to know how often the number one would come up. The probability is defined as follows:

Probability = (# of favourable results) divided by (# of possible results)

A “favourable result” is the result for which you are determining the probability (i.e. the probability of rolling the number 1)

Since there are 6 sides to a die, there are 6 possible results.

Therefore, the probability of rolling a 1 would be 1/6. This can be expressed as a decimal and a percentage: 1/6 = .17 = 17%

Probability In The Real World:

For events such as flipping a coin or rolling a die, we can identify all the possible outcomes, and we can assume that (1) the outcome on any one event is determined by chance and (2) all outcomes are equally probable.

For these events that are not so easily analyzed, we can use a different method to determine probability: conduct a series of tests, or “trials”, of the event we are interested in and observe the outcomes. Once a sufficiently large number of trials have been conducted, we determine the probability of the event in question with the following formula:

Probability = Number of favourable outcomes/total number of trials

Suppose we want to determine the probability that a basketball player will make a basket from the foul line. Since we cannot assume that all outcomes are equally probable or that any one outcome will be determined by chance, we cannot determine the probability using only a formula: we have to conduct a number of trials.

Suppose we observe the player taking 1,000 consecutive foul shots, and he makes 700 of them. We now have the data necessary to estimate probability.

700 (number of favourable outcomes) divided by 1,000 (total number of trials) = 70%

Therefore it is determined that for each time this particular player goes to the foul line to take a shot, the probability that he will make it is 70%. Unlike probability based on a mathematical formula, probability based on trials should be thought of as an estimate: it is based on previous occurrences, and there is no guarantee that future trials will follow the same pattern.

The reason for our uncertainty is that these outcomes are not random: that is, they are not based on chance. If the player was to practice foul shots every day for a month and we were to conduct a number of trials again, the outcome might be different than it was the first time.

Combined Probability

  • When we deal with more than one event, probability becomes more complicated.

Instead of flipping a single coin, suppose you were flipping two coins and wanted to determine the probability of both coins coming up heads. In other words, you want to know the probability of the first coin coming up heads and the second coin coming up heads. The probability that both events will occur is the product of the two separate probabilities.

Probability of the 1st and 2nd event occurring = (Probability of 1st) x (Probability of 2nd)

When flipping two coins, the probability of both coming up heads is .5 multiplied by .5, or .25.

For example, the probability of the Basketball Player making two shots in a row is: (.7)(.7) = .49 or just about 50/50.

Probability of Dependent and Independent Events

  • Independent events:
    The outcome of the 1st event DOES NOT affect the outcome of the 2nd event.
  • Dependent events:
    The outcome of the 1st event DOES affect the outcome of the second event.

Example of independent events: If you are flipping a coin twice, the two flips are independent: the outcome of the first flip does not affect the outcome of the second in any way.

Example of dependent events: You want to determine the probability of drawing an ace from a well-shuffled deck (i.e. the cards are randomly placed). Since there are 4 aces and 52 cards in all, the probability of drawing an ace on your first try is 4/52 or 1/13 (or approximately 7.7%). If you do not draw an ace on your first try, your chances of drawing an ace on your second try are no longer 4 out of 52: there are still 4 aces in the deck, but you have drawn a card out, so there are only 51 remaining cards to choose from. Thus, as a result of the previous outcome, your chances of drawing an ace have changed to 4/51, or approximately 7.8%. If you had drawn an ace on your first try, your chances of drawing an ace on your second try would be 3/51, or approximately 5.9%. In this card drawing example, each trial removes one of the outcomes from the pool of possible outcomes; once you have drawn a card, that card is no longer in the pool of possible outcomes for future trials. Since each trial affects the potential outcomes of subsequent trials, these events are said to be dependent.

How does this relate to gambling? It is easy to mistake independent events for dependent events. This is known as the “Gambler’s Fallacy”.

The gambler’s fallacy reflects the belief that because a coin has come up heads several times in a row, it is more likely to come up tails on the next flip (Paulos, 1988, p.43). In reality, the outcomes are totally independent: the coin does not “remember” what it has done in the past in order to “decide” what it will do next. Or, said differently, an outcome of heads on one trial does not remove heads from the pool of possible outcomes for subsequent trials. No matter what has happened before, the probability of tails for any one coin toss is always 50%.