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Mutually exclusive events are events that cannot occur simultaneously. When one event happens, it rules out the possibility of the other events taking place. For example, when you toss a coin, the events 'getting heads' and 'getting tails' are mutually exclusive. You can either get heads or tails, but never both.
Mathematically, this concept is represented by the intersection of the two events being an empty set. So, if events \( E \) and \( F \) are mutually exclusive, we write \( E \cap F = \emptyset \). This means there are no outcomes that are common to both events.

Probability theory is a branch of mathematics that deals with the likelihood of different outcomes. It helps us understand how likely events are to happen. Probabilities are always between 0 and 1, where 0 means an event will not happen and 1 means it certainly will happen.
To find the probability of mutually exclusive events happening at the same time, we need to consider their intersection. Since disjoint events have no common outcomes, the probability of both happening at the same time is zero. This is written as \( P(E \cap F) = 0 \).
Key points to remember include:

• The sum of probabilities of all possible outcomes is always 1.
• The probability of an impossible event is 0.
• The probability of a certain event is 1.

The intersection of events in probability theory refers to the outcomes that are common to both events. For events \( E \) and \( F \), the intersection is written as \( E \cap F \).
When events are disjoint or mutually exclusive, they have no common outcomes. This means their intersection is an empty set (\( \emptyset \)). Consequently, the probability of the intersection of disjoint events is always zero. Hence, \( P(E \cap F) = 0 \) for disjoint events.
Understanding intersections is important because it helps us calculate the likelihood of multiple events happening together. If events are not mutually exclusive, their intersection would have some common outcomes, and we would need to account for this when calculating probabilities.