Independent and Dependent Events

In our page about Probability Theory we briefly explained that the probability of an event occurring (say, rolling a three on a dice), is equal to the number of ways that that event can happen (in this case, one), divided by the total number of possible events in the given scenario (in this case, six). So, the probability of rolling a three is one divided by six, or 0.1667. The ways of working out the probability of an event depend heavily on whether or not the events are independent or dependent – but what exactly is meant by this?

The difference between independent and dependent events

An independent event is one whose outcome is not based on the outcome of another event, and one whose outcome does not affect the outcome of another event, whereas a dependent event is the opposite of an independent event in that its outcome does affect, or is affected by, the outcome of another event.

As an example of independent events: if we have two dice (see the mathematics of craps for more info on dice mathematics) and we roll them both, the value that is rolled on the first dice is not influenced by the value that is rolled on the second dice, and the first dice’s value also does not play a role in determining the value on the second dice. Another example of independent events is two throws of the same coin – the probability of the second throw landing on heads or tails is always 0.5 (in other words, 1 in 2), regardless of whether the first throw landed on heads or tails. The mathematics of poker largely involve dependant events, since the number of cards from which to draw decreases every time a card is dealt, meaning that the probability of drawing a certain card from the deck increases for each time that that card is not dealt.

The probability of two independent events occurring at the same time (dice) or consecutively (two throws of one coin) is the probability of those events multiplied with each other. So with the dice, where the probability of throwing any value is 0.1667, we can work out that the probability of, for example, throwing a 3 on the one dice and a 4 on the other is 0.1667 x 0.1667 = 0.027. With the coins, the probability of throwing a heads and then a tails would be 0.5 x 0.5 = 0.25 (i.e. a one in four chance).

An example of a scenario where independent events apply in gambling is the lottery. People often think that if they stick with a certain set of numbers for long enough, the chances of those numbers coming up improve with each time that they don’t come up. This is not true, as each lottery draw is an independent event – that is, previous lottery draws have no influence on subsequent lottery draws. Thus, if you’re planning on winning the lottery, your chances are the same regardless of whether you choose new numbers each week or if you stick with the same set of numbers!

Visit our gambling mathematics homepage for more information on the mathematic principles that govern all forms of casino games, or view our gambling mathematics glossary.