What are Independent Events?
Independent events in statistics are those in which one event does not affect the next event. More specifically, the occurrence of one event does not affect the probability of the following event happening.
Here are three quick examples of independent events:
- Flipping a coin. The outcome of one coin toss has no impact on the results of the following toss.
- Rolling a die. One outcome does not impact the next die roll.
- The probabilities of rain and mail delivery are independent events. Mail delivery occurs or not, regardless of the rain.
Have you ever felt that you are “overdue” for an event to happen? For example, if you have been flipping a coin and it has come up tails several times in a row, you might feel that you are “due” for it to come up heads. Unfortunately, this feeling is inaccurate in the context of independent events.
Each coin flip is an independent event. The outcome of one flip does not affect the result of the next. If you have gotten tails four times in a row, the probability of tails on the next flip is still 50/50. Independent events have no “memory” of previous events.
The gambler’s fallacy is a cognitive bias that occurs in the context of independent events. It is that sense of being “overdue” for a favorable outcome given a previous series of unfavorable results. However, probability theory indicates that is incorrect.
Probability of Independent Events
Calculating the probability of independent events is straightforward. For one event, you simply divide the number of ways an event can happen by the total number of outcomes.
For example, the probability of rolling a 6 on a fair six-sided die is 1/6 because there is only one favorable outcome (rolling a 6) out of six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6).
When you have two or more independent events, you can use the specific multiplication rule to calculate joint probabilities. Calculate the likelihood of both events happening by multiplying the probability of each event happening individually. For example, suppose you want to know the chances of rolling two sixes on a pair of dice. The chances of rolling a six on the first die is 1/6. The likelihood of rolling a six on the second die is also 1/6 because the die rolls are independent events.
Consequently, the joint probability of rolling two sixes in a row is 1/6 x 1/6 = 1/36.
You can use the specific multiplication rule for finding the chances of sharing a birthday in the classic Birthday Problem. When you have dependent events, you must use the general multiplication rule. Learn more in-depth about the specific and general forms of the rule in Multiplication Rule for Calculating Probabilities.
One of the fascinating things about independent events is that the chances of getting many heads (or tails) in a row is relatively low. For example, the likelihood of flipping a coin and getting heads is 1/2. The probability of getting two successive heads is 1/2 x 1/2 = 1/4. The chance of getting three consecutive heads is 1/2 x 1/2 x 1/2 = 1/8. As you can see, the probability gets smaller and smaller with each additional head. Consequently, getting many heads in a row is rare, even though each flip has a 1/2 chance of landing on heads.
Example of Calculating Probability: Rolling Two Dice
Let’s take a closer look at an example of independent events involving rolling two dice. Suppose you roll two dice. What is the probability that you roll a sum of 7?
To solve this problem, we need to determine the number of ways we can roll a sum of 7. There are six ways to roll a 7: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). Because each die has six possible outcomes, there are 6 x 6 = 36 total possible outcomes.
Therefore, the probability of rolling a sum of 7 is 6/36 or 1/6. Notice that the outcome of the first die roll has no impact on the result of the second die roll, so these are independent events.
In conclusion, understanding independent events is crucial to calculating probabilities in statistics. By knowing when events are independent, you can easily calculate the likelihood of multiple occurrences by multiplying the individual probabilities. Remember, the occurrence of one event does not impact the chances of the next one happening.
When you have independent events, you can use probability distributions like the binomial and negative binomial distributions. These distributions allow you to calculate probabilities for the number of occurrences and when the first event will occur, respectively.
For a more advanced look at independent events, I recommend considering them from the perspective of conditionality, which you can read about in my post Conditional Probabilities: Definition, Formula & Examples.