The Mathematics of Gambling: What is the difference between dependent and independent events?

The difference between dependent and independent events is a crucial one to understand if one is to be able to accurately assess probabilities and have a fair idea of ones chances of winning or losing at any particular game.

The notion of an “independent event” is often misunderstood by people. The term “independent event” is a term used in probability theory to refer to past events that have no influence on future events. For example, if you flip a coin five times the probability of getting five heads in a row is:


The probability of five heads in a row is 0.03125 because the probability of getting a head when you flip the coin once is 0.5 and the flipping of a coin is an example of an independent event. This means that the results of previous flips do not influence future flips. No matter how many time you flip the coin, or whether the previous flips resulted in heads or tails, for each and every flip the probability of getting a head remains 0.5. Therefore we calculate the probability of getting five heads in a row like this:

0.5 x 0.5 x 0.5 x 0.5 x 0.5 = 0.03125

Many people misunderstand the reality of probabilities and independent events, and somehow believe that if a coin is flipped four times and lands on heads each time, it is more likely that on the next flip the coin will land on tails – that the coin is somehow “due” to land on tails. Although they do not articulate this thought, many people believe that the past flips will influence the future flips. The reality is that each and every time you flip the coin, the odds of getting heads or tails will remain identical – 0.5.

You will often see similar thought patterns and behaviours when people are picking numbers for a lottery. People generally tend to have “their numbers” - the numbers they prefer to choose every time they buy a ticket. They tend to feel that the more times the lottery draw produces numbers that are not the ones they have chosen, then the more likely it is that their numbers will come up in the next draw. You will also find that if someone has been choosing the same numbers for a very long time and then is lucky enough to win with those numbers, they will change the numbers they choose after that. They feel that it is extremely unlikely that the same sequence of numbers will come up again. In fact, every single sequence of numbers is just as likely (or rather, unlikely) to be drawn as every other, despite whether or not it has been drawn before. You will also know, or maybe have experienced yourself, the feeling of “almost” winning – of getting “close” to winning. If you have picked the number 17 to be drawn and the number 18 is drawn, you will feel that you “nearly won”, that you came close to winning. In fact, 17 and 18 are as different in the context of lottery drawings as 17 and 199. There is no “almost winning” or “coming close to winning” - either your number is drawn, or a number that is not yours is drawn. You win or you lose. Feeling that sense of “nearly” winning is just another example of people misunderstanding the nature of independent events.

Playing the pokies is kind of like flipping a coin, in that each spin is an independent event. The odds remain the same for every single spin, no matter what may have happened in previous spins. A machine is never “due” to pay out, the same way a coin is never “due” to come up heads or tails.

There are some games you can play, however, in which you operate with dependent events. Naturally then, if an independent event is an event which is not influenced by events which have come before it, and which does not influence events to come, a dependent event is the opposite of this: an event whose odds or probability changes depending on the events which have preceded it; and even whose outcome will alter the probability of events to come. Many card games need to be played while considering the nature of dependent events, if one is to have an accurate understanding or the probabilities one is playing with. Consider a deck of cards. If the cards are not replaced into the deck after they have been dealt, then the dealing of each hand is a dependent event: the probability of getting any one particular card varies according to which cards have already been removed from the deck. For example, for the first card to be drawn from the full deck, the probability of getting a King is 4/52 or 1/13 or approximately 0.076923. If this card is not replaced however, then the probability that the second person will also be dealt a king has changed. There are now only three Kings remaining in the deck, so the chances are 3/52 or approximately 0.057692. This applies to all card games. Keeping track of the changing probabilities in a game involving dependent events is much more difficult than playing with independent events. However, you have a greater chance of winning in these games, particularly if you can keep track of the changing probabilities. There are different ways of doing this. Very talented poker players can hold in their mind all the cards which have been dealt previously, and therefore can calculate the cards which must either remain in the deck or be in the hands of other players. Being able to roughly (or precisely, depending on the skill of the player) calculate the probability of a player having a particular hand can be used to great advantage. There are simpler ways to keep track of changing probabilities as well though, which do not require superior mathematical ability or memory. Blackjack players often use a simple method of counting cards, which allows them to have a rough idea of whether they or the dealer has a greater chance of winning a particular hand. See Frequently Asked Question What is card counting? for more details on this and instructions on how to go about it.