A compound event is an event that has more than one possible outcome. In a compound event, an experiment gives more than one possible outcome. These outcomes may have different probabilities but they are all equally possible. In this sub-topic, we will see what we mean by compound events.

An event is an occurrence that can be determined by a given level of certainty. A compound event combines two or more events, using the word “AND” or the word “OR”. Probability of a compound event can be written as an expression involving probabilities of simpler events. By doing this, calculation of compound probability becomes easier.

Mutually exclusive events

Those events which cannot happen simultaneously are called mutually exclusive events. In other words, two events are mutually exclusive if they cannot occur at the same time. For two mutually exclusive events E₁ and E₂, E₁ ∩ E₂ = ∅. For example, in a throw of die, we cannot get an even and odd outcome at the same time. Thus, getting an even number or odd number in a throw of a die are mutually exclusive events.

The “OR” rule: For mutually exclusive events E₁ and E₂.
P (E₁ or E₂) = P (E₁) + P (E₂)

Non-mutually exclusive events

If there is at least one outcome which is common in the events, they are called non-mutually exclusive events. In other words, if two events have sample points in common, they are not mutually exclusive. For example, in drawing a card event of drawing a spade and an ace card are not mutually exclusive event because a spade can also be an ace card.

Independent events

Independent events are events where the occurrence of one of the events does not affect the occurrence of the other event.
Consider an experiment of tossing a coin and throwing a die simultaneously. To determine the probability of getting a head on a coin and a "5" on a dice simultaneously, we have to combine the outcomes of the coin and the dice. We know that the outcome for a coin is either head or tail. Outcomes for a die can be either 1, 2, 3, 4, 5 or 6. By combining these outcomes, we will have the following sample space:

{(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)}

Here, (H, 1) represents head on the coin and 1 on the dice.

Dependent events

Dependent events are events where the occurrence of one of the events does affect the occurrence of the other event. In other words, dependent events are events for which the occurrence of either event affects the next.

If E₁ and E₂ are dependent events, then P (E₁ then E₂) = P(E₁) × P (E₂) when E₁ has occurred).