Difference Between Mutually Exclusive and Independent Events

Mutually Exclusive vs Independent Events

In mathematics, the probability between two events bears some characteristics like mutuality, exclusivity, and dependency. These concepts are all very tricky, but upon learning by example, these probability concepts are actually very simple. Take, for example, the difference between mutually exclusive and independent events. At first glance, the two terms seem the same, but, in fact, they are very different.

“Independent events” means that the probability (pr) of two events (event x and event y) are not affected or independent from each other. In mathematical notation, the pr (x and y) = pr (x) . pr (y). The probability that the two events (x and y) will happen is equal to the likelihood that “x” happens multiplied by the likelihood that “y” happens.

In a mutually exclusive case, the scenario becomes different. Using the same variables as above, the pr (x and y) = 0. This means that the likelihood of event “x” and “y” occurring altogether or at the same time is absolutely zero. This also means that the two events are not independent of each other and, therefore, they are mutually exclusive. In simpler terms, this would mean that if event “x” is present, event “y” will surely not happen.

Here are some tangible examples of the two situations above. In independent events using the variables “x” and “y,” variable “x” represents obtaining tails in a simple coin toss, and “y” represents obtaining ”1” from a die toss. Using the formula on independent events, the equation is pr (x and y) = pr (x) . pr (y) = 1/2 . 1/6 = 1/12. Clearly, the product is not equal to zero.

Using the same toss coin example, “x” now represents obtaining heads while “y” represents obtaining tails. Although the likelihood of getting a heads and tails are both 1 out of 2, still these events are mutually exclusive because getting heads and tails at the same time with one coin toss is not possible. With this it is safe to say that two, mutually exclusive events are dependent events, the presence or occurrence of one affects the presence or occurrence of the other.

Summary:

1.“Independent events” means that the occurrence or outcome of one event does not influence the occurrence of another event.
2.“Mutually exclusive” events means that the occurrence or presence of one event entails the non-occurrence of the other.
3.Independent events are expressed mathematically as pr (x and y) = pr (x) . pr (y) while mutually exclusive events are expressed as pr (x and y) = 0.

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