If A and B are two mutually exclusive events, then what is the probability of occurrence of either event A or event B?
P(A) + P(B)
Understanding Mutually Exclusive Events Probability
The question asks for the probability of occurrence of either event A or event B when A and B are two mutually exclusive events. In probability, the word "or" typically corresponds to the union of events, denoted by the symbol \( \cup \). So, we are looking for \( P(A \cup B) \).
What are Mutually Exclusive Events?
Mutually exclusive events (also known as disjoint events) are events that cannot happen at the same time. If event A occurs, event B cannot occur, and vice versa. There is no overlap between the two events. In terms of set theory, the intersection of two mutually exclusive events A and B is the empty set \( \emptyset \), meaning \( A \cap B = \emptyset \).
Probability of the Union of Two Events
The general formula for the probability of the union of two events A and B (the probability that either A or B occurs, or both) is given by the Addition Rule:
\( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
Here, \( P(A) \) is the probability of event A, \( P(B) \) is the probability of event B, and \( P(A \cap B) \) is the probability that both event A and event B occur (the probability of their intersection).
Applying the Formula to Mutually Exclusive Events
For mutually exclusive events, we know that the intersection \( A \cap B \) is impossible. Therefore, the probability of their intersection is 0:
\( P(A \cap B) = P(\emptyset) = 0 \)
Substituting this into the general Addition Rule formula:
\( P(A \cup B) = P(A) + P(B) - 0 \)
\( P(A \cup B) = P(A) + P(B) \)
Thus, the probability of the occurrence of either event A or event B, when they are mutually exclusive events, is simply the sum of their individual probabilities.
Conclusion on Probability of A or B
Based on the definition of mutually exclusive events and the rules of probability, the probability of either event A or event B occurring is \( P(A) + P(B) \).
Revision Table: Probability Concepts
Mutually Exclusive Events: Events that cannot happen at the same time (\( A \cap B = \emptyset \)).
Probability of Union (A or B): General formula is \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).
Probability of Union for Mutually Exclusive Events: \( P(A \cup B) = P(A) + P(B) \) because \( P(A \cap B) = 0 \).
Intersection (A and B): Denoted by \( A \cap B \). Probability is \( P(A \cap B) \).
Additional Information: Joint Probability
The term \( P(A \cap B) \) is often referred to as the joint probability of events A and B. It represents the probability that both events A and B occur simultaneously. For mutually exclusive events, the joint probability is always zero, as they cannot occur together. Understanding the difference between the union (\( \cup \), "or") and the intersection (\( \cap \), "and") of events is fundamental in probability theory, especially when dealing with concepts like independence versus mutual exclusivity.
Similar Questions
What is \(P(\overline T | \overline G)\) equal to?
What is \(P(G | \overline T) \) equal to?
What is \(P (G \cap \overline T)\) equal to?
Let two events A and B be such that P(A) = L and P(B) = M. Which one of the following is correct?
If P(A|B) < P(A), then which one of the following is correct?
If A and B are two events such that P(not A) = \(\rm \frac{7}{10}\) , P(not B) = \(\rm \frac{3}{10}\) and P(A|B) = \(\rm \frac{3}{14}\) , then what is P(B|A) equal to?
A and B are two events such that A̅ and B̅ are mutually exclusive. If P(A) = 0.5 and P(B) = 0.6, then what is the value of P(A|B)?
Three groups of children contain 3 girls and 1 boy; 2 girls and 2 boys: 1 girl and 3 boys. One child is selected at random from each group. The probability that the three selected consist of 1 girl and 2 boys is
Two integers x and y are chosen with replacement from the set (0, 1, 2…10). The probability that |x - y| > 5 is
If two dice are thrown and at least one the dice show 5, then the probability that the sum is 10 or more is
Important Questions from Conditional Probability
If P(A) = 0.7, P(B) = 0.5 and P(B/A) = 0.3, find (i) P(A/B) (ii) P(A ∪ B)?
Let A and B be two events such that \(P(\overline{A \cup B}) = \dfrac{1}{6}\) , P(A ∩ B) = \(\dfrac{1}{4}\) and P(A̅) = \(\dfrac{1}{4}\) , where A̅ stands for complement of event A. Then, events A and B are:
A bike manufacturing factory has two plants P and Q. Plant P manufactures 60 percent of bikes and plant Q manufacture 40 percent. 80 percent of the bikes at plant P and 90 percent of the bikes at plant Q are rated of standard quality. A bike is chosen at random and is found to be of standard quality. What is the probability that it has come from plant P?
20 percent of the pens produced in a factory are of red colour and 4 percent are red and defective. If one pen is picked up at random, then what is the probability of its being defective if it is red?
In a game, there are three rooms- I, Il and IIl. Room I contain 2 boxes having gift items and 3 empty boxes, room II contains 3 boxes having gift items and 2 empty boxes, and room III contains 4 boxes having gift items and one empty box respectively. There is an equal probability of each room being chosen by a player. Mr John selects one box from a room chosen at random. The probability that Mr John wins a box having gift items is:
