I. \((A \cup B \cup C) \cap (\overline{A} \cap \overline{B} \cap \overline{C})\) is an impossible event.
II. \((A \cap B \cap C) \cap (\overline{A} \cup \overline{B} \cup \overline{C})\) is a possible event.
Which of the statements given above is/are correct ?
Analyzing Statement I: Impossible Event
Statement I considers the expression \((A \cup B \cup C) \cap (\overline{A} \cap \overline{B} \cap \overline{C})\).
- Let \(X = A \cup B \cup C\). This represents the event where at least one of A, B, or C occurs.
- By De Morgan's laws, \(\overline{A} \cap \overline{B} \cap \overline{C} = \overline{A \cup B \cup C}\). This represents the event where none of A, B, or C occur.
- The expression becomes \((A \cup B \cup C) \cap (\overline{A \cup B \cup C})\).
- An event and its complement cannot happen simultaneously. The intersection of any event \(E\) and its complement \(\overline{E}\) is always the impossible event, denoted by \(\emptyset\).
- Therefore, \((A \cup B \cup C) \cap (\overline{A \cup B \cup C}) = \emptyset\). Statement I is correct.
Analyzing Statement II: Possible Event
Statement II considers the expression \((A \cap B \cap C) \cap (\overline{A} \cup \overline{B} \cup \overline{C})\).
- Let \(P = A \cap B \cap C\). This represents the event where all of A, B, and C occur.
- By De Morgan's laws, \(\overline{A} \cup \overline{B} \cup \overline{C} = \overline{A \cap B \cap C}\). This represents the event where it is not the case that all of A, B, and C occur.
- The expression becomes \((A \cap B \cap C) \cap (\overline{A \cap B \cap C})\).
- Similar to Statement I, the intersection of an event and its complement is the impossible event, \(\emptyset\).
- Thus, \((A \cap B \cap C) \cap (\overline{A \cap B \cap C}) = \emptyset\). This is an impossible event.
- Statement II claims this is a possible event, which is false. Statement II is incorrect.
Conclusion
Only Statement I is correct.
Similar Questions
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Important Questions from Probability
The probability of being 53 Sundays in year 2020 is-
Three dice are thrown randomly. The probability of coming 3 in at least one die is
The probability of having 53 Tuesdays in an ordinary year is:
Two dice are thrown simultaneously The probability of getting the sum 2 or 8 or 12 is
If two balanced dice are tossed once, the probability of the event, that the sum of the integers coming on the upper sides of the two dice is 9, is
