Mathematics: Gate 2005 EE

Question

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)

Comments

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.

Answer 1

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 $\cap$ B) =  P(A)* P(B)  so option A is false

B) P(A $\cup$ B) =  P(A) + P(B) - P(A $\cap$ B)

hence P(A $\cup$ B) >= P(A) + P(B)  its false option B is false

C) if A and B is mutually exclusive means P(A $\cap$ B) =  0 then its independent ........its false  there is no relation independent and mutually exclusive

D)  P(A $\cap$ B) <= P(A)

   its true

hence option D is correct

Comments

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

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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

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Yeah, this helped.

To be more precise, I think this should be like

Probability (P ∩ Q) ≤ min{ Probability (P), Probability (Q) }

Answer 2

Let's analyse options one by one.

(a)  Independence of P and Q implies that probability (P ∩ Q) = 0

   No. This is the case for mutually exclusive events that if one occurs then the chance of another one occuring is ruled out.

  Infact Independence of P and Q means if P happens, then outcome of Q won't be affected by that.

  Means  P(Q|P) =Q because it has not effect whether P happens or not

which makes P(P $\cap$ Q) =P(P).P(Q) because again P(P|Q)= P.

(b) Probability (P ∪ Q) ≥ Probability (P) + Probability (Q)

This can hold true for mutually exclusive events P and Q but if P and Q can happen together(implies independence)

 then P(P $\cap$ Q) $\neq$ 0.

So P( P U Q )=  P(P) + P(Q) - P(P $\cap$ Q)

So, in this case P(PUQ) $\leq$ P(P) +P(Q)

(c) If P and Q are mutually exclusive, then they must be independent.

Absolutely false. When P and Q are mutually exclusive, then either P occurs or Q occurs but not both simultaneously. So if P happens, chance of Q happening gets ruled out and vice-versa. Hence, mutually exclusive events are dependent.

(d) Probability (P ∩ Q) ≤ Probability (P)

Consider 2 cases

Case 1 : P and Q are mutually exclusive.

Let S be sample space of outcomes of a single toss of a coin.

Let P : Outcome is head

   Q : Outcome is tails.

 Obviously, P(P $\cap$ Q) =0 and P(P)= $\frac{1}{2}$

Equality satisfied.

Case 2: P and Q are independent.

P : Outcome of getting head on a single toss of a coin

Q : Outcome of getting 4 on dice.

P(P $\cap$ Q) = $\frac{1}{12}$

P(P)=$\frac{1}{2}$

Equality satisfied.

Option (d) TRUE.

Answer 3

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 ...

Answer 4

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