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
- What is the probability that getting a total of the numbers on the dice is 6 ?
- What is the probability that getting a total of the numbers on the dice is at least 23 ?
- In a sample survey of a village, the probability that a farmer is in debt is 0.60. What is the probability that three randomly selected farmers are all in debt (assume independence of events)?
- The probability that a family owns a laptop is 0.68; that it also owns a desktop is 0.56. If the probability that it owns both is 0.48, then what is the probability that a randomly selected family owns a laptop or a desktop?
Important Questions from Probability
Two dice are thrown simultaneously and the sum of the numbers appearing on them is noted. What is the probability that the sum is 12?
If a box contains 3 white cushions, 4 red cushions and 5 blue cushions, what is the probability of selecting a white or blue cushion?
A. 2/3
B. 3/4
C. 1/4
D. 1/9Statements followed by some conclusions are given below.
Statements:
1. A bag has 2 white, 3 black, 4 red and 6 green balls.
2. 1 ball selected at random from the bag.
Conclusions:
I. The probability that a black ball is selected is 1/5
II. The probability that a red ball is selected is 6/15
Find which of the conclusions logically follows from the given statement
A. Only conclusion I follows.
B. Only conclusion II follows.
C. Both I and II follow.
D. Neither I nor II follows.
In a shooting test, the probabilities of hitting the target are 1/2 for A, 2/3 for B and 3/4 for C. If they fire at the same target, what is the probability that only one of them hits the target?
A bag contains balls numbered from 1 to 42. One ball is drawn at random from these balls. The probability that its number is a multiple of 7 or 8 is:
