When calculating the probability of the occurrence of one event or another event of two mutually exclusive events which method is appropriate?

The addition rule for probabilities describes two formulas, one for the probability for either of two mutually exclusive events happening and the other for the probability of two non-mutually exclusive events happening.

Índice Show

What is a Mutually Exclusive Event?
Mutually Exclusive Event Probability
Mutually Exclusive Event PRobability: Steps

The first formula is just the sum of the probabilities of the two events. The second formula is the sum of the probabilities of the two events minus the probability that both will occur.

The addition rule for probabilities consists of two rules or formulas, with one that accommodates two mutually-exclusive events and another that accommodates two non-mutually exclusive events.
Non-mutually-exclusive means that some overlap exists between the two events in question and the formula compensates for this by subtracting the probability of the overlap, P(Y and Z), from the sum of the probabilities of Y and Z.
In theory the first form of the rule is a special case of the second form.

Mathematically, the probability of two mutually exclusive events is denoted by:

$\text{ or } Z) = P(Y)+P(Z)$

Mathematically, the probability of two non-mutually exclusive events is denoted by:

$\text{ or } Z) = P(Y) + P(Z) - P(Y \text{ and } Z)$

To illustrate the first rule in the addition rule for probabilities, consider a die with six sides and the chances of rolling either a 3 or a 6. Since the chances of rolling a 3 are 1 in 6 and the chances of rolling a 6 are also 1 in 6, the chance of rolling either a 3 or a 6 is:

1/6 + 1/6 = 2/6 = 1/3

To illustrate the second rule, consider a class in which there are 9 boys and 11 girls. At the end of the term, 5 girls and 4 boys receive a grade of B. If a student is selected by chance, what are the odds that the student will be either a girl or a B student? Since the chances of selecting a girl are 11 in 20, the chances of selecting a B student are 9 in 20 and the chances of selecting a girl who is a B student are 5/20, the chances of picking a girl or a B student are:

11/20 + 9/20 - 5/20 =15/20 = 3/4

In reality, the two rules simplify to just one rule, the second one. That's because in the first case, the probability of two mutually exclusive events both happening is 0. In the example with the die, it's impossible to roll both a 3 and a 6 on one roll of a single die. So the two events are mutually exclusive.

Mutually exclusive is a statistical term describing two or more events that cannot coincide. It is commonly used to describe a situation where the occurrence of one outcome supersedes the other. For a basic example, consider the rolling of dice. You cannot roll both a five and a three simultaneously on a single die. Furthermore, getting a three on an initial roll has no impact on whether or not a subsequent roll yields a five. All rolls of a die are independent events.

Probability and Statistics > Probability > Mutually Exclusive Events

What is a Mutually Exclusive Event?

Mutually exclusive events are things that can’t happen at the same time. For example, you can’t run backwards and forwards at the same time. The events “running forward” and “running backwards” are mutually exclusive. Tossing a coin can also give you this type of event. You can’t toss a coin and get both a heads and tails. So “tossing a heads” and “tossing a tails” are mutually exclusive. Some more examples are: your ability to pay your rent if you don’t get paid, or watching TV if you don’t have a TV.

A coin toss can be mutually exclusive.

Watch the video for the definition and two examples of finding probabilities for mutually exclusive events:

Mutually Exclusive Events

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Mutually Exclusive Event Probability

The basic probability(P) of an event happening (forgetting mutual exclusivity for a moment) is:
P = Number of ways the event can happen / total number of outcomes.
Example: The probability of rolling a “5” when you throw a die is 1/6 because there is one “5” on a die and six possible outcomes. If we call the probability of rolling a 5 “Event A”, then the equation is: P(A) = Number of ways the event can happen / total number of outcomes

P(A) = 1 / 6.

It’s impossible to roll a 5 and a 6 together; the events are mutually exclusive.

The events are written like this:

P(A and B) = 0
In English, all that means the probability of event A (rolling a 5) and event B (rolling a 6) happening together is 0.

However, when you roll a die, you can roll a 5 OR a 6 (the odds are 1 out of 6 for each event) and the sum of either event happening is the sum of both probabilities. In probability, it’s written like this:

P(A or B) = P(A) + P(B)
P(rolling a 5 or rolling a 6) = P(rolling a 5) + P(rolling a 6)
P(rolling a 5 or rolling a 6) = 1/6 + 1/6 = 2/6 = 1/3.

It’s impossible to roll a 1 and a 2 together.

Mutually Exclusive Event PRobability: Steps

Example problem: “If P(A) = 0.20, P(B) = 0.35 and (P A∪ B) = 0.51, are A and B mutually exclusive?”

Note: a union (∪) of two events occurring means that A or B occurs.

Step 1: Add up the probabilities of the separate events (A and B). In the above example:
.20 + .35 = .55

Step 2: Compare your answer to the given “union” statement (A ∪ B). If they are the same, the events are mutually exclusive. If they are different, they are not mutually exclusive. Why? If they are mutually exclusive (they can’t occur together), then the (∪)nion of the two events must be the sum of both, i.e. 0.20 + 0.35 = 0.55.

In our example, 0.55 does not equal 0.51, so the events are not mutually exclusive.

Like the explanation? Check out more step by step examples—just like this one for mutually exclusive events—in the Practically Cheating Statistics Handbook

References

Beyer, W. H. CRC Standard Mathematical Tables, 31st ed. Boca Raton, FL: CRC Press, pp. 536 and 571, 2002.
Dodge, Y. (2008). The Concise Encyclopedia of Statistics. Springer.
Everitt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics, Cambridge University Press.
Wheelan, C. (2014). Naked Statistics. W. W. Norton & Company

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