A and B are any two events such that P(A ∪B)=(5)/(6), P(A ∩B)=(1)/(3), P(B)=(1)/(2), then A and

step 1 So in the given question we are told that A and B are events such that the probability of A unio

A and B are any two events such that P(A ∪B)=(5)/(6), P(A...

$\mathrm{A}$ and $\mathrm{B}$ are any two events such that $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{5}{6}, \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{3}, \mathrm{P}(\mathrm{B})=\frac{1}{2}$, then $\mathrm{A}$ and $\mathrm{B}$ are (a) equally likely (b) exhaustive (c) mutually exclusive (d) independent

$\mathrm{A}$ and $\mathrm{B}$ are any two events such that $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{5}{6}, \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{3}, \mathrm{P}(\mathrm{B})=\frac{1}{2}$, then $\mathrm{A}$ and $\mathrm{B}$ are
(a) equally likely
(b) exhaustive
(c) mutually exclusive
(d) independent

Key Concepts

Equally Likely Events

Events are equally likely if every event in a set of outcomes has the same probability of occurring. This concept simplifies analysis by allowing assumptions of uniform probability distribution over the events, a common starting point in basic probability problems.

Mutually Exclusive Events

Events are mutually exclusive if they cannot occur at the same time; that is, their intersection is empty (or has probability zero). This concept is important because if events are mutually exclusive, it is impossible for both to occur simultaneously, contrasting with the idea of independence.

Exhaustive Events

Exhaustive events cover all possible outcomes in a sample space. This means that at least one of the events must occur, ensuring the total probability over the sample space is one. This concept is fundamental when partitioning the whole sample space into distinct events.

Inclusion-Exclusion Principle

This principle is used to compute the probability of the union of two events by adding the probabilities of each event and then subtracting the probability of their intersection. It helps address any overlap between events so that probabilities are not counted twice.

Independent Events

Two events are independent if the probability of their intersection is the product of their individual probabilities, meaning that the occurrence of one event does not affect the probability of the other. This concept is critical in probability theory as it allows for simplified computations when events do not influence each other.

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