Correct Answer - Option 2 : Independent.
Concept:
For two events A and B, we have: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
- If A and B are independent events, then P(A ∩ B) = P(A) × P(B).
- If A and B are mutually exclusive events, then P(A ∩ B) = 0.
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\(\rm P(\bar A)=1-P(A)\).
Calculation:
Since \(\rm P(\bar{B}) = \dfrac{1}{2}\), therefore \(\rm P(B)=1-P(\bar{B}) = 1-\dfrac{1}{2}=\dfrac{1}{2}\).
Using the relation P(A ∪ B) = P(A) + P(B) - P(A ∩ B), we get:
\(\rm \dfrac56=P(A)+\dfrac12-\dfrac13\)
⇒ \(\rm P(A)=\dfrac56-\dfrac16=\dfrac46=\dfrac23\)
We can observe that \(\rm P(A)\times P(B)=\dfrac23\times\dfrac12=\dfrac13=P(A\cap B)\).
∴ A and B are independent events.
- A mutually exclusive event is defined as a situation where two events cannot occur at same time.
When a coin is a tossed, there are two events possible, either it will be a Head or a Tail. Both the events here are mutually exclusive because they cannot happen simultaneously.
- An independent event is where one event remains unaffected by the occurrence of the other event.
If we take two separate coins and flip them, then the occurrence of a Head or a Tail on both the coins are independent of each other, because a Head/Tail on one coin, does not affect the outcome of the other coin.
- Mutually exclusive events are necessarily also dependent events because one's existence depends on the other's non-existence.
If at least one of the events has zero probability, then the two events can be mutually exclusive and independent simultaneously. However, if both events have non-zero probability, then they cannot be mutually exclusive and independent simultaneously.
