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Given that the events A and B are such that P(A) = 1/2, P (A ∪ B) = 3/5, and P(B) = p. Find p if they are (i) mutually exclusive (ii) independent
Welcome to AskiitiansGiven, P(A) = 1/2 ,P (A ∪ B) = 3/5and P(B) = p.(1) For Mutually ExclusiveGiven that, the sets A and B are mutually exclusive.Thus, they do not have any common elementsTherefore, P(A ∩ B) = 0We know that P(A ∪ B) = P(A) + P(B) – P(A ∩ B)Substitute the formulas in the above-given formula, we get3/5 = (1/2) + p – 0Simplify the expression, we get(3/5) – (1/2) = p(6 − 5)/10 = p1/10 = pTherefore, p = 1/10Hence, the value of p is 1/10, if they are mutually exclusive.(ii) For Independent events:If the two events A & B are independent,we can write it as P(A ∩ B) = P(A) P(B)Substitute the values,= (1/2) × p= p/2Now, P(A ∪ B) = P(A) + P(B) – P(A ∩ B)Now, substitute the values in the formula,(3/5) = (1/2)+ p – (p/2)(3/2)– (1/2)= p – (p/2)(6 − 5)/10 = p/21/10 = p/2p= 2/10P = 1/5Thus, the value of p is 1/5, if they are independent.Thanks
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