Aspire Faculty ID #3925 · Topic: NIMCET 2019 · Just now
NIMCET 2019

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?

Solution

Statement 1 is false: 
Two events can have equal probability without being identical. 

 Statement 2 is false: 
If $P(U)=0$, it only means $U$ is a null-probability event, not that $U^{c}=S$. 

 Statement 3 is incomplete, so it cannot be true. 

 Statement 4 is true: 
 If $U$ and $V$ are independent,
then $P(U \cap V) = P(U)P(V)$
 We check independence of complements: 
 $P(U^{c} \cap V^{c}) = 1 - P(U) - P(V) + P(U \cap V)$ 
 $= 1 - P(U) - P(V) + P(U)P(V)$ 
 $= (1 - P(U))(1 - P(V))$ 
 $= P(U^{c}) P(V^{c})$ 
 Hence, $U^{c}$ and $V^{c}$ are independent.
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Prince Pathak
Prince Pathak , Dropper
Commented Feb 21, 2024
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