Question

For three events A, B, C, P(Exactly one of A and B occurs)=P (Exactly one of B and C occurs)=P(exactly one of C or A occurs)= “\[\dfrac{1}{4}\]” and P(All the three events occur simultaneously)= “\[\dfrac{1}{{16}}\]”, Then the probability that at least one of the events occur is:
A. \[\dfrac{7}{{32}}\]
B. \[\dfrac{7}{{16}}\]
C. \[\dfrac{7}{{64}}\]
D. \[\dfrac{3}{{16}}\]

Answer 1

Hint: The given question had explained the relation between the probabilities of the three events, to solve such question you have to use the property of probability which explains that for any three or more events then by using their probabilities we can solve them as per given condition, no any extra formulae is needed.

Complete step by step solution:
For the given question conditions are given for three events, lets derive the mathematical form for every condition and then solved for the question, for that we have to assume some variables according to the demand of the question and then derive the expression from the statement given in the question, on solving we get:
\[
   \Rightarrow \text{exactly one of A and B occurs} = x \\
   \Rightarrow \text{exactly one of B and C occurs}= y \\
   \Rightarrow \text{exactly one of C and A occurs} = z \\
   \Rightarrow \text{all three events occurs simultaneously} = k \\
\text{ Now according to question}: \\
   \Rightarrow {P_x} = {P_y} = \dfrac{1}{4} = {P_z},{P_k} = \dfrac{1}{{16}} \\
   \Rightarrow {P_x} = P(A \cap B),{P_y} = P(B \cap C),{P_z} = P(C \cap A) \\
 \]
We know that-
\[
   \Rightarrow P(A \cup B \cup C) = P(A \cap B) + P(B \cap C) + P(C \cap A) - P(A \cap B \cap C) \\
   \Rightarrow P(A \cap B \cap C) = - P({P_x} \cup {P_y} \cup {P_z}) + \left[ {P(A \cap B) + P(B \cap C) + P(C \cap A)} \right] \\
   \Rightarrow P(A \cap B \cap C) = - P({P_k}) + \left[ {\dfrac{1}{4} + \dfrac{1}{4} + \dfrac{1}{4}} \right] \\
 \]

\[
   \Rightarrow P(A \cap B \cap C) = - \dfrac{1}{{16}} + \dfrac{1}{4} \\
   \Rightarrow P(A \cap B \cap C) = \dfrac{3}{{16}} \\
 \]

So, the correct answer is “Option D”.

Note: To solve the question of probability you need to see the condition given in the question, the theorem available in probability can be used only for fixed questions, in which range of data is needed, otherwise you cannot solve for the given question. Here for the above question we take help of set theory because of the limitations of data.