A shop sells cigars of brand A as well as brand B. Let X be the event

Question

A shop sells cigars of brand A as well as brand B, among other brands. Let X be the event that a customer buys at least one cigar of brand A, let Y be the event that a customer buys at least one cigar of brand B, and let Z be the event that a customer buys cigar of neither brand A nor brand B. Probability of event X is denoted as P(X), probability of event Y is denoted as P(Y), probability of event Z is denoted as P(Z). What is the probability of X U Y, P(XUY) i.e., what is the probability of either event X or event Y.

(1) P(Z) = 0.35.

(2) P(X) = 0.40 and P(Y) = 0.38.

(1) P(Z) = 0.35.

(2) P(X) = 0.40 and P(Y) = 0.38.

Archit3110 · Answer

#1:

p(XuY) = p(X) + p(Y) - p(XnY)
or
p(XuY) = 1 - p(XnY)' = 1 - Probability of neither of the events occurring = 1 - p(Z)
 P(Z) = 0.35 - Sufficient

#2:
P(x)=P(a)=0.4 and P (y)=P(b)=0.38
P(a)+P(b)= 0.4+.38 = 0.78; not sufficient

IMO A

Diwakar003 · Answer

p(XuY) = p(X) + p(Y) - p(XnY)
or
p(XuY) = 1 - p(XnY)' = 1 - Probability of neither of the events occurring = 1 - p(Z)

(1) P(Z) = 0.35 - Sufficient

(2) P(X) = 0.40 and P(Y) = 0.38 - We don't know p(XnY) - Not sufficient.

Hence I'll go with A.

Cheers!

egmat · Answer

Events (X ∪ Y) and Z are  complementary  - they cover ALL possibilities with no overlap.

Think of it this way:
• A customer either buys at least one cigar from A or B → this is event (X ∪ Y)
• OR they buy neither A nor B → this is event Z
• There's no third option!

Therefore: P(X ∪ Y) + P(Z) = 1, which means  P(X ∪ Y) = 1 - P(Z)

Statement 1:  P(Z) =  0.35 
Using our relationship:
P(X ∪ Y) = 1 - 0.35 =  0.65

We get one definite answer.   SUFFICIENT

Statement 2:  P(X) =  0.40  and P(Y) =  0.38 
The formula for union is: P(X ∪ Y) = P(X) + P(Y) - P(X ∩ Y)

P(X ∪ Y) = 0.40 + 0.38 - P(X ∩ Y) = 0.78 - P(X ∩ Y)

Problem: We don't know P(X ∩ Y) - the probability that a customer buys BOTH brand A AND brand B.

Case 1:  If no one buys both brands, P(X ∩ Y) = 0 → P(X ∪ Y) = 0.78
 Case 2:  If some customers buy both, say P(X ∩ Y) = 0.13 → P(X ∪ Y) = 0.65

Different values possible.   NOT SUFFICIENT

Answer: A

A shop sells cigars of brand A as well as brand B. Let X be the event

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A shop sells cigars of brand A as well as brand B. Let X be the event

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