The intersection of two events E₁ and E₂, denoted by E₁ ∩ E₂, is the event consisting of all outcomes that are in both E₁ and E₂. Sometimes E₁ and E₂ have no outcomes in common so that the intersection of A and B contains no outcomes. In this sub-topic, we will study the probability of events joined by the word “AND”, in other words, the intersection of events.

If two events E₁ and E₂ are associated with “AND”, then it means the intersection of elements which are common to both events. The intersection symbol “∩” is used to represent “AND” in probability.

Thus, the events E₁ and E₂ denotes E₁ ∩ E₂.

The intersection of two events is the probability that the two events, E₁ and E₂, will occur at the same time. If both events are mutually exclusive, then this probability will be 0 because both events cannot occur at the same time.

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 either1, 2, 3, 4, 5 or 6. By combining these outcomes, we will have the following sample space as follows:

(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.

The “AND” rule: If E₁ and E₂ are independent events, then

P E₁ and E₂) = P(E₁) × P(E₂).