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.
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Important Questions from Probability
Two dice are thrown simultaneously. What is the probability of getting the same number on both the dice?
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:
When two dice are tossed simultaneously, the probability that the sum of the numbers appearing on both the dice is 8 will be
