Given two fair dice, what is the probability that the sum of their num

Question

Given two fair dice, what is the probability that the sum of their numbers is 4 if exactly one die shows a 3?

(A) 2/11
(B) 1/18
(C) 3/11
(D) 2/39
(E) 1/5

Kalli1 · Answer

2 possible combinations 3+1 , 1+3 . Total combinations 6^2 . So 2/36 = 1/18 , Correct B

BAA5 · Answer

Bunuel   chetan2u   VeritasKarishma  could you please post the solution
My understanding was this 
Given that exactly 1 of the dice shows 3, the other dice can have 1,2,4,5,6 as possible results, the probability that sum is 4 would then be probability of getting 1 out of the 5 possible events= 1/5

or it could be the probability of getting 3 on first dice and 1 on second dice = 1/6*1/5 or 1 on first dice, 3 on second dice 1/5*1/6 which would be 1/15 but since getting 3 is given I opted for answer E

chetan2u · Answer

I would agree to answer E.

EXACTLY is the catch word. 
If exactly one is 3, the other has to be 1. So (1,3) and (3,1) are 2 ways.

However, total ways are 
First dice 3, the other can be remaining 5 = 5 ways.
First anything other than 3 and second dice 3 = 5 ways.
So total 10 ways.

P= 2/10=1/5

If the question said AT LEAST one is 3, then the answer would be the given OA 2/11 as both dice showing 3 will also add up.

KarishmaB · Answer

You are given that one die shows a 3. It means the other die does not show a 3. So total cases would be those in which exactly one die shows a 3. 
Assuming the dice to be distinct (say Red and Yellow), we can get a sum of 4 in 2 ways (3 on Red, 1 on Yellow or 3 on Yellow and 1 on Red).
The total possible ways with a 3 on exactly one die are 5 (3 on Red and 1/2/4/5/6 on Yellow) + 5 (3 on Yellow and 1/2/4/5/6 on Red) = 10

Probability = 1/5

Vibhatu · Answer

Given that   exactly   1 of the dice shows 3, the other dice can have 1,2,4,5,6 
So there are two possible ways, (3,1) or (1,3)
then probability: 1/5 *1/5= 1/5 (E)
 ANSWER: E

yashhw · Answer

Let me define the events:
 
 A = Sum of the two dice is 4 
 B = Exactly one die shows a 3

We want  P(A | B) .

Step 1: Find P(A ∩ B) 
Outcomes where the sum is 4 AND exactly one die shows a 3:
 
 only (3, 1) and (1, 3)  
So  P(A ∩ B) = 2/36

Step 2: Find P(B) 
Outcomes where exactly one die shows a 3:
 
 Die 1 = 3, Die 2 ≠ 3 → 5 outcomes: (3,1), (3,2), (3,4), (3,5), (3,6) 
 Die 2 = 3, Die 1 ≠ 3 → 5 outcomes: (1,3), (2,3), (4,3), (5,3), (6,3)  
So  P(B) = 10/36

Step 3: Apply Bayes' Theorem (Conditional Probability) 
P(A∣B)=P(A∩B)/P(B)=1/5

Given two fair dice, what is the probability that the sum of their num

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Given two fair dice, what is the probability that the sum of their num

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