Let U and V be two events of a sample space S and P(A) denote the probability of an event A. Which of the following statements is true?
If P(U) = 0, then Uc = S.
Events U and V Probability Rules
This question requires identifying the correct statement concerning probability and events within a sample space S.
Probability Statements Analysis
- Option 1: If P(U) = P(V), then U = V.
This statement is not always true. Different events can have the same probability value without being the same event. For example, if you roll a fair six-sided die, the event of rolling a '1' and the event of rolling a '2' both have a probability of P = 1/6, but they are distinct events. - Option 2: If P(U) = 0, then Uc = S.
This statement is true. The probability of an event U, P(U), represents its likelihood. If P(U) = 0, it signifies an impossible event. The complement of U, denoted Uc, includes all outcomes not in U. According to the complement rule, P(Uc) = 1 - P(U). Given P(U) = 0, this means P(Uc) = 1 - 0 = 1. The only event with a probability of 1 is the entire sample space, S. Therefore, Uc must be equal to S. - Option 3: If U ∩ V = φ, then U and V are independent.
This statement is not necessarily true. Events where U ∩ V = φ are called mutually exclusive. For events to be independent, the condition P(U ∩ V) = P(U)P(V) must hold. If events are mutually exclusive, P(U ∩ V) = P(φ) = 0. This equation (0 = P(U)P(V)) is only true if P(U) or P(V) (or both) is 0. If their probabilities are non-zero, mutually exclusive events are dependent. - Option 4: If U and V are independent, then so are Uc and Vc.
This statement is also true based on probability theory principles.
The focus is on explaining the true statement identified.
P(U)=0 Rule Explanation
This section details why the statement "If P(U) = 0, then Uc = S" is correct:
- Understanding Zero Probability: An event U having P(U) = 0 implies it's an impossible event or has zero measure in the context of the sample space S. This means the event U does not contain any outcomes, or any outcomes it contains have zero probability.
- The Complement Rule: The probability of the complement of an event U, written as P(Uc), is calculated using the fundamental formula P(Uc) = 1 - P(U).
- Applying the Rule: When the probability of event U is given as 0 (P(U) = 0), we substitute this value into the complement rule. This yields P(Uc) = 1 - 0, which simplifies to P(Uc) = 1.
- Probability of the Sample Space: By definition in probability theory, the probability of the entire sample space S occurring is always 1. This is often written as P(S) = 1. The sample space represents the set of all possible outcomes.
- Logical Conclusion: Since we calculated that the probability of the complement event Uc is 1 (P(Uc) = 1), this directly implies that the event Uc must encompass all possible outcomes within the sample space. Therefore, Uc is equivalent to the sample space S.
Important Questions from Probability of Random Experiments
Let A, B and C be three mutually exclusive and exhaustive events associated with a random experiment. If P (B) = 1.5 P (A) and P (C) = 0.5 P (B), then P (A) is equal to
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A problem in Mathematics is given to 3 students A, B and C. If the probability of A solving the problem is \(\dfrac{1}{2}\) and B not solving it is \(\dfrac{1}{4}\) and the whole probability of the problem being solved is \(\dfrac{63}{64}\), then what is the probability of solving it by C?
