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Mutually exclusive events are ones that cannot happen at the same time. This is like saying you cannot be in two places at once.
For example, when you roll a die, you cannot get both an even and an odd number. Mutually exclusive events may also be called disjoint events.
To put it simply: if one event happens, the other one cannot happen.

Probability is a measure of how likely an event is to happen. It is often written as a number between 0 and 1.
A probability of 0 means the event will never happen, while a probability of 1 means it is certain to happen.
For instance, the probability of rolling a '4' on a fair six-sided die is \( \frac{1}{6} \).
When dealing with disjoint events, the probability of both occurring together is always 0. This helps in better predicting possible outcomes.

The formal definition of disjoint or mutually exclusive events is when two events, say A and B, have no outcomes in common.
In mathematical language, this is written as:
\[ P(A \text{ and } B) = 0 \]
Here, \( P(A \text{ and } B) \) represents the probability of both events occurring at the same time. Since they are disjoint, their joint probability is 0.
This means there is no overlap between the events. If one happens, the other cannot.

To understand disjoint events better, think of a standard six-sided die. Let's define two events:

• Event A: Rolling an even number (2, 4, 6)
• Event B: Rolling an odd number (1, 3, 5)

These events are disjoint because a single roll cannot result in both an even and an odd number.
We can further explore the implications: If events A and B are disjoint, \( \text{P(A or B)} \) is the sum of their individual probabilities:
\tex \[ \text{P(A or B)} = \text{P(A)} + \text{P(B)}\]Let's say each outcome of a die roll has a probability of \( \frac{1}{6} \). Then:
\[ \text{P(A)} = \text{P(2 or 4 or 6)} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{1}{2} \]
\[ \text{P(B)} = \text{P(1 or 3 or 5)} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{1}{2} \]
Therefore, \[ \text{P(A or B)} = \frac{1}{2} + \frac{1}{2} = 1 \]
This ensures that one of the two events will surely occur.