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
- An edible oil is sold at the rates 150, 200, 250, 300 rupees per litre in four consecutive years. Assuming that an equal amount of money is spent on oil by a family in every year during these years, what is the average price of oil in rupees (approximately) per litre ?
- Let \(m = 77^n\). The index \(n\) is given a positive integral value at random. What is the probability that the value of \(m\) will have 1 in the units place ?
- What is \(P(A) + P(B) + P(C)\) equal to ?
- A, B, C are three mutually exclusive and exhaustive events associated with a random experiment. If \(3P(B) = 4P(A)\) and \(3P(C) = 2P(B)\), then what is \(P(A)\) equal to?
- A die has two faces with number 4, three faces with number 5 and one face with number 6. If the die is rolled once, then what is the probability of getting 4 or 5?
- A box contains 2 black, 4 yellow and 6 white balls. Three balls are drawn in succession with replacement. What is the probability that all three are of the same colour?
- The letters of the word ZOOLOGY are arranged in all possible ways. What is the probability that the consonants and vowels occur alternatively?
- A natural number \(x\) is chosen at random from the first 100 natural numbers. What is the probability that \(x^2 + x > 50\)?
- A fair coin is tossed till two heads occur in succession. What is the probability that the number of tosses required is less than 6?
- Two perfect dice are thrown. What is the probability that the sum of the numbers on the faces is neither 9 nor 10?
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:
When two dice are tossed simultaneously, the probability that the sum of the numbers appearing on both the dice is 8 will be
How many 3 - digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?
