Question

If A and B are two events such that $ P(A) = \dfrac{1}{2} $ and $ P(B) = \dfrac{2}{3} $ then which of the following is correct?
 $ (1)\;{\text{P(A}} \cup {\text{B)}} \geqslant \dfrac{2}{3} $
 $ (2)\;{\text{P(A}} \cap {\text{B) }} \leqslant \dfrac{1}{3} $
 $ (3) $ Both (1) & (2)
 $ (4) $ Only 1

Answer 1

Hint: Here we will solve one by one for the intersection of the two events and the union of the two events by following the condition of the union and the intersection and will select the correct option. Here Union means maximum value which can go and the intersection is the minimum value which can go for the probability of the two events

Complete step-by-step answer:
First, the union for the probability of the two events is the maximum value between the two probabilities and it can be greater than that.
 $ P(A) = \dfrac{1}{2} $ and $ P(B) = \dfrac{2}{3} $
 $ \Rightarrow P(A \cup B) \geqslant \dfrac{2}{3} $ ..... (i)
Similarly, the probability of the intersection of the two events is the minimum probability among the two probabilities or it may be less than that also.
 $ P(A) = \dfrac{1}{2} $ and $ P(B) = \dfrac{2}{3} $
 $ \Rightarrow P(A \cap B) \leqslant \dfrac{1}{3} $ ..... (ii)
Hence, from the given multiple choices – the option C is the correct answer.
So, the correct answer is “Option C”.

Note: If the given two events A and B are mutually exclusive then the intersection of both the events is equal to zero. The intersection of the two events is the set of the elements which belongs to both the sets and is denoted by - $ \cap $ . Also, the probability of the intersection of the Events A and B is represented by - $ P(A \cap B) $ . Whereas, the union of the two sets are the set of all the elements in A and B and is denoted by- $ \cup $ . Also, the probability of the intersection of the Events A and V is represented by - $ P(A \cup B) $ .