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Here you will find answers to your questions on the topic stochastic independence. This article covers the Independence from events based on an illustrative Example. We also calculate the probabilities with the associated one formula.
Our Video on the topic explains briefly everything you need to know about Independence from events should know without having to read this article!
Independence from events
The stochastic independence of events implies that the occurrence of one event has no effect on the likelihood of occurrence of the other event. The event A is called stochastically independent of the event B if the probability P (A) is not influenced by it. It does not matter whether the second event occurs or not.
For example, the likelihood that someone will have blue eyes is not related to the likelihood that that person will pass the statistics exam.
Stochastic independence formula
As can be seen, mathematically expressed for independent events, the following applies:
The conditional probability so corresponds exactly to that unconditional probability.
That is also logical, since the occurrence of B by definition has no influence on the occurrence of A and vice versa. With this assumption, the probability can be calculated with this formula:
But be careful! This formula can only be used for independent events. If the events are dependent, you have to use the following formula:
Stochastic independence tasks
In order to solve tasks related to stochastic independence, you can also use various aids. With the help of this one can map the given information in a structured way. This makes the calculation easier afterwards. A simple four-field board or a Venn diagram enable a better overview of the task with little effort.
Independence in the tree diagram
A tree diagram is also ideal for thisIndependence from events to illustrate. This would look like this, for example:
Stochastic independence example
Now let's look at a suitable example on the subject. Imagine a die is thrown once. We define “odd number” as event A and “number less than 5” as event B. Now you should determine whether the events A and B are different from each other dependent or independent are.
Calculate stochastic independence
First we need to determine the probability of the two events. Since the event A comprises three elements and the result B four, there is a probability of in each case or. .
Next, we have to think about how many elements there are at the intersection of A and B, i.e. how many elements appear in both A and B. These are the numbers 1 and 3.
Accordingly, the intersection of A and B results in a probability of .
Check stochastic independence
Now we can simply use the formula from before to check whether the events are interdependent or not. The following must apply to independent events:
In our case:
The events A and B are thus statistically independent of one another.
Stochastic and causal dependency
In conclusion, it is important to point out that stochastic dependency is not the same as causal dependency that you may know from your everyday life.
Two events can be stochastically dependent, even if they are not related to each other in cause and effect. Here are the formulas that are important in connection with independent events:
The following applies to independent events:
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