Final Answer:
The probability of the union of two events A and B is equal to the sum of the individual probabilities minus the intersection of the two events.
Explanation:
The probability of an event A or B is calculated using the equation P(A or B) = P(A) + P(B) – P(A ∩ B). This equation can be derived from the principle of inclusion and exclusion. The inclusion-exclusion principle states that when two events occur together, the probability of either event is equal to the sum of the probabilities of the individual events minus the probability of their intersection.
For example, let's say we have two events A and B. The probability of event A is P(A) and the probability of event B is P(B). The probability of the union of the events is the sum of the individual probabilities, P(A) + P(B). However, the probability of the intersection of the two events is also taken into account, as it would be double counted in the sum. Thus, the probability of the union of the two events is equal to the sum of the individual probabilities minus the intersection of the two events, P(A or B) = P(A) + P(B) – P(A ∩ B).
This equation can be extended to more than two events. If there are n events, the probability of the union is equal to the sum of the individual probabilities minus the sum of the probabilities of all possible intersections of the events. Thus, the probability of the union of n events is equal to the sum of the individual probabilities minus the sum of the probabilities of all possible intersections of the events, P(A or B or C or ... or n) = P(A) + P(B) + P(C) + ... + P(n) – P(A ∩ B) – P(A ∩ C) – ... – P(B ∩ C) – ... – P(n ∩ n).
In the given equation, P(A'B') = 1 - P(A) - P(B) + P(A ∩ B), A' and B' represent the complement of events A and B, respectively. The probability of the complement of an event is equal to one minus the probability of the event. Thus, the probability of the complement of A is P(A') = 1 – P(A), and the probability of the complement of B is P(B') = 1 – P(B). The probability of the union of the complements of A and B is equal to the sum of the probabilities of the complements minus the intersection of the complements. Thus, P(A'B') = 1 - P(A) - P(B) + P(A ∩ B).
In conclusion, the probability of the union of two events A and B is equal to the sum of the individual probabilities minus the intersection of the two events. This equation can be extended to more than two events, as the probability of the union of n events is equal to the sum of the individual probabilities minus the sum of the probabilities of all possible intersections of the events. Using the complement of an event, the equation can be simplified to P(A'B') = 1 - P(A) - P(B) + P(A ∩ B).
Question:
What is the formula for calculating the probability of the union of two events A and B, how is it derived from the principle of inclusion and exclusion, and how can this formula be extended to more than two events? Furthermore, explain the role of complements in simplifying the equation, as mentioned in the provided explanation.