Let A and B be two events such that P(A ∪B)=(1)/(6), P(A ∩B)=(1)/(4) and P(A̅)=(1)/(4), where A̅

step 1 We know that P probability of A union B bar can be represented as 1 minus probability of A union

Let A and B be two events such that P(A ∪B)=(1)/(6), P(A...

Let $A$ and $B$ be two events such that $P(\overline{A \cup B})=\frac{1}{6}, P(A \cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{1}{4}$, where $\bar{A}$ stands for complement of event $A$. Then events $A$ and $B$ are (A) equally likely and mutually exclusive (B) equally likely but not independent (C) independent but not equally likely (D) mutually exclusive and independent

Let $A$ and $B$ be two events such that $P(\overline{A \cup B})=\frac{1}{6}, P(A \cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{1}{4}$, where
$\bar{A}$ stands for complement of event $A$. Then events $A$ and $B$ are
(A) equally likely and mutually exclusive
(B) equally likely but not independent
(C) independent but not equally likely
(D) mutually exclusive and independent

Key Concepts

Independent Events

Independent events are defined as events where the occurrence of one does not influence the occurrence of the other. This relationship is characterized by the condition P(A ? B) = P(A) × P(B). Understanding independence is essential for determining how probabilities combine in scenarios without interaction between events.

Equally Likely Events

Equally likely events are those that have the same probability of occurring. Recognizing when events are equally likely can simplify probability calculations and is important for understanding symmetry in probability problems.

Mutually Exclusive Events

Mutually exclusive events are those that cannot occur simultaneously, meaning the occurrence of one event precludes the occurrence of the other, which is mathematically expressed as P(A ? B) = 0. This concept is key when evaluating scenarios where events do not overlap.

Complement of an Event

The complement of an event consists of all outcomes in the sample space that are not in the event. Its probability is given by P(Complement) = 1 - P(Event) and is fundamental in calculating the likelihood of an event not occurring, especially when direct computation is complex.

Union and Intersection of Events

The union of two events represents all outcomes that belong to at least one of the events, while the intersection consists of the outcomes that are common to both events. The inclusion-exclusion principle, which states that P(A ? B) = P(A) + P(B) - P(A ? B), is crucial in avoiding the double counting of outcomes in overlapping events.

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