Question

Let A and B be two events such that ?

\(P(\overline{A\cup B}) = \dfrac{1}{6}, P(A\cap B) = \dfrac{1}{4} \ \text{and} \ P(̅{A}) = \dfrac{1}{4}\) where A̅ stands for complement of event A. Then the events A and B are

1. independent but not equally likely
2. mutually exclusive and independent
3. equally likely and mutually excllisive
4. equally likely but not independent

Answer 1

Correct Answer - Option 1 : independent but not equally likely

Concept:

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

The probability of the complement of an event is one minus the probability of the event. P (A̅) = 1 - P (A)

To determine the probability of two independent events we multiply the probability of the first event by the probability of the second event. P(A ∩ B) = P(A).P(B)

Two events are mutually exclusive when two events cannot happen at the same time. P(A ∩ B) = 0

In equally likely events, the probabilities of each event are equal.

 

Calculation:

Here, \(P(\overline{A\cup B}) = \dfrac{1}{6}, P(A∩ B) = \dfrac{1}{4} \ \text{and} \ P(̅{A}) = \dfrac{1}{4}\)

P(A) = 1 - P(A̅) = 1 - 1/4 = 3/4

\(\rm P(A\cup B)=1-P(\overline{A\cup B}) = 1-\dfrac{1}{6}=\frac 56\)

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

⇒ P(B) = 5/6 - 3/4 + 1/4 = 5/6 - 1/2 

= 2/6

P(B) = 1/3

Here, P(A ∩ B) = P(A).P(B) and P(A) ≠ P(A) so the events are independent but not equally likely.

Hence, option (1) is correct.