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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)
asked in Probability by Loyal (4.4k points) 5 17 33 | 237 views

Although many people(@Hradesh patel, @Tendua, @Kantikumar and @Vicky rix) have given correct answers but i think following statement is not looking proper -->

there is no relation independent and mutually exclusive

Refer --> 

https://math.stackexchange.com/questions/941150/what-is-the-difference-between-independent-and-mutually-exclusive-events

https://math.stackexchange.com/questions/2469209/can-two-events-be-mutually-exclusive-but-not-independent/2469213

In above mentioned reference observe the following line -->

This of course means mutually exclusive events are not independent, and independent events cannot be mutually exclusive.

 

2 Answers

+3 votes
Best answer

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(AÜ B) >= 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

 

 

answered by Boss (7.1k points) 5 40 149
selected by

First three options are false for sure but still not satisfied with last one. Can you please elaborate more, @Hradesh patel ? 

u think this way   A ^ B  <=  A     ////  in this line" < "denotes subset

                         n(A  ^  B ) <=  n(A)

                         P(A ^ B) <=  P(A)

i hope this help u
Yeah, this helped.

To be more precise, I think this should be like

Probability (P ∩ Q) ≤ min{ Probability (P), Probability (Q) }
0 votes
A) Let random experiment be tossing 2 coins ... Let E1 be probability of getting a head in first coin and E2 be probability of getting       tails in 2 nd coin ... Here both E1 and E2 are independent but P(E1 AND E2) = 1/4 != 0 ..so A) is eliminated ...

B) P(P OR Q) = P(P) + P(Q) WHEN P and Q are mutually exclusive . Else P(P OR Q) < P(P) + P(Q) .So P(P OR Q) <= P(P) +P(Q) ... So B) is FALSE ..

C) There is no relation between mutually exclusive and independent events ...

D) is TRUE always ... Ex : Probability of getting good rank in gate >= probability of getting good rank in gate AND getting into iits ...
answered by Loyal (4.9k points) 3 35 121


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