Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > If P and Q are two random events, then the fo...
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If P and Q are two random events, then the following is TRUE:
- a)Independence of P and Q implies that probability (P ∩ Q) = 0
- b)Probability (P ∪ Q) ≥ Probability (P) + Probability (Q)
- c)If P and Q are mutually exclusive, then they must be independent
- d)Probability (P ∩ Q) ≤ Probability (P)
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
If P and Q are two random events, then the following is TRUE:a)Indepen...
here i check option---> i let P = probability, P and Q is replaced by A and B.
A) it is saying independent means P(A ^ B) = P(A)* P(B) so option A is false
B) P(AÜ B) = P(A) + P(B) - P(A ^ B)
hence P(AÜB) >= P(A) + P(B) its false option B is false
C) if A and B is mutually exclusive means P(A ^ B) = 0 then its independent ........its false there is no relation independent and mutually exclusive
D) P(A ^ B) <= P(A)
A) it is saying independent means P(A ^ B) = P(A)* P(B) so option A is false
B) P(AÜ B) = P(A) + P(B) - P(A ^ B)
hence P(AÜB) >= P(A) + P(B) its false option B is false
C) if A and B is mutually exclusive means P(A ^ B) = 0 then its independent ........its false there is no relation independent and mutually exclusive
D) P(A ^ B) <= P(A)
its true
hence option D is correct
Most Upvoted Answer
If P and Q are two random events, then the following is TRUE:a)Indepen...
here i check option---> i let P = probability, P and Q is replaced by A and B.
A) it is saying independent means P(A ^ B) = P(A)* P(B) so option A is false
B) P(AÜ B) = P(A) + P(B) - P(A ^ B)
hence P(AÜB) >= P(A) + P(B) its false option B is false
C) if A and B is mutually exclusive means P(A ^ B) = 0 then its independent ........its false there is no relation independent and mutually exclusive
D) P(A ^ B) <= P(A)
A) it is saying independent means P(A ^ B) = P(A)* P(B) so option A is false
B) P(AÜ B) = P(A) + P(B) - P(A ^ B)
hence P(AÜB) >= P(A) + P(B) its false option B is false
C) if A and B is mutually exclusive means P(A ^ B) = 0 then its independent ........its false there is no relation independent and mutually exclusive
D) P(A ^ B) <= P(A)
its true
hence option D is correct
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Community Answer
If P and Q are two random events, then the following is TRUE:a)Indepen...
Probability (P ∩ Q) ≤ Probability (P)
To understand why option D is the correct answer, let's break down the statement and discuss each part separately.
Probability (P ∩ Q) represents the probability of both events P and Q occurring simultaneously. This can also be written as the joint probability of P and Q.
Probability (P) represents the probability of event P occurring.
The statement in option D states that Probability (P ∩ Q) ≤ Probability (P).
To understand why this statement is true, consider the following:
1. If events P and Q are independent, then the occurrence of one event does not affect the probability of the other event occurring. In this case, the joint probability of P and Q would be equal to the product of the probabilities of each event occurring independently.
For example, if the probability of event P occurring is 0.5 and the probability of event Q occurring is 0.3, then the joint probability of P and Q occurring would be 0.5 * 0.3 = 0.15.
In this case, the joint probability (0.15) is less than or equal to the probability of event P occurring (0.5), which confirms the statement in option D.
2. If events P and Q are dependent, then the occurrence of one event affects the probability of the other event occurring. In this case, the joint probability of P and Q would not necessarily be less than or equal to the probability of event P occurring.
For example, if the probability of event P occurring is 0.5 and the probability of event Q occurring is 0.7, and there is a strong dependence between the two events, the joint probability of P and Q occurring could be greater than 0.5.
In this case, the joint probability would not satisfy the inequality in option D, indicating that the events are dependent.
Therefore, the statement in option D holds true because it covers both scenarios of independent and dependent events. If events P and Q are independent, the joint probability would be less than or equal to the probability of event P occurring. If events P and Q are dependent, the joint probability could be greater than the probability of event P occurring, thus not satisfying the inequality.
To understand why option D is the correct answer, let's break down the statement and discuss each part separately.
Probability (P ∩ Q) represents the probability of both events P and Q occurring simultaneously. This can also be written as the joint probability of P and Q.
Probability (P) represents the probability of event P occurring.
The statement in option D states that Probability (P ∩ Q) ≤ Probability (P).
To understand why this statement is true, consider the following:
1. If events P and Q are independent, then the occurrence of one event does not affect the probability of the other event occurring. In this case, the joint probability of P and Q would be equal to the product of the probabilities of each event occurring independently.
For example, if the probability of event P occurring is 0.5 and the probability of event Q occurring is 0.3, then the joint probability of P and Q occurring would be 0.5 * 0.3 = 0.15.
In this case, the joint probability (0.15) is less than or equal to the probability of event P occurring (0.5), which confirms the statement in option D.
2. If events P and Q are dependent, then the occurrence of one event affects the probability of the other event occurring. In this case, the joint probability of P and Q would not necessarily be less than or equal to the probability of event P occurring.
For example, if the probability of event P occurring is 0.5 and the probability of event Q occurring is 0.7, and there is a strong dependence between the two events, the joint probability of P and Q occurring could be greater than 0.5.
In this case, the joint probability would not satisfy the inequality in option D, indicating that the events are dependent.
Therefore, the statement in option D holds true because it covers both scenarios of independent and dependent events. If events P and Q are independent, the joint probability would be less than or equal to the probability of event P occurring. If events P and Q are dependent, the joint probability could be greater than the probability of event P occurring, thus not satisfying the inequality.
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If P and Q are two random events, then the following is TRUE:a)Independence of P and Q implies that probability (P ∩ Q) = 0b)Probability (P ∪ Q) ≥ Probability (P) + Probability (Q)c)If P and Q are mutually exclusive, then they must be independentd)Probability (P ∩ Q) ≤ Probability (P)Correct answer is option 'D'. Can you explain this answer?
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