Question of the Week- 20 (A biased dice is such that any odd number..)

Question

e-GMAT          Question of the Week #20

A biased dice is such that any odd number appears thrice as frequently as any even number. If the dice is rolled twice, what is the probability that the sum of the numbers thus obtained is 10?

A.  1/16
 B.  5/72
 C.  11/144
 D.  11/72
 E.  11/16

https://e-gmat.com/blogs/wp-content/uploads/2018/08/Quant-workshop_Webinar-e1533628475738.jpg

GMATinsight · Accepted Answer

HAPPYatHARVARD

probability of 4 = Probability of getting even number * probability of getting 4 out of 3 even numbers {2,4,6}  
Now probability of getting an even number on this dice = 1/4
Probability of getting 4 out of three even numbers {2,4,6} = 1/3
i.e. Probability of getting 4 on unbiased dice = 1/4*1/3

I hope this helps

GMATinsight · Answer

If Probability of even Number = x
then Probability of odd number = 3x

But, Probability of even Number on dice + Probability of odd Number on dice = 1

i.e. x+3x = 1
i.e. x = 1/4

Sum may be 10 in following ways
(4,6) (5,5) (6,4)

Probability of (4,6) =  Probability of 4  followed by  probability of 6  =  (1/3)*(1/4)  (1/3)*(1/4)  = 1/144

Probability of (6,4) =  Probability of 6  followed by  probability of 4  =  (1/3)*(1/4)  (1/3)*(1/4)  = 1/144

Probability of (5,5) =  Probability of 5  followed by  probability of 5  =  (1/3)*(3/4)  (1/3)*(3/4)  = 9/144

Probability for the entire desired outcome = (1/144)+(1/144)+(9/144) = (11/144)

Answer: Option C

Please follow color coding to understand respective probabilities.

EgmatQuantExpert  I think it would have been better to mention that the dice still has six faces 3 of which are odd as the Dice may be biased on these parameters as well

HAPPYatHARVARD · Answer

GMATinsight  
Hi,
I am not able to understand why you multiplied 1/3 .
ex: probability of 4 = 1/4*1/3
Thanks

gvij2017 · Answer

Please elaborate why did you multiply all probabilities with 1/3?

GMATinsight · Answer

gvij2017

Rationale for using the probabilities is as follows:

Probability of getting an even number on dice = 1/4
Probability of getting an Odd number on dice = 3/4

Probability of getting a specific even number e.g. 2 out of three even numbers {2,4,6} = 1/3
Probability of getting a specific odd number e.g. 3 out of three odd numbers {1,3,5} = 1/3

In each case either we calculate probability of a specific even number or an odd number therefore in each case we have used 1/3 for that specific even number or odd number.

I hope this helps!!!

EgmatQuantExpert · Answer

Solution 
  Given:  
 • In a biased dice, any odd number appears thrice as frequently as any even number

To find:  
 • The probability that the sum of the numbers obtained upon rolling the dice twice is 10

Approach and Working:   
 • Let us assume that the frequency of getting any even number, {2, 4, 6} is x
 o Then, the frequency of getting any odd number, {1, 3, 5} will be 3x

• Implies, 
 o The probability of getting any even number =  \frac{x}{(x + 3x + x + 3x + x + 3x)} = \frac{1}{12} 
o And, the probability of getting any odd number =  \frac{3x}{(x + 3x + x + 3x + x + 3x)} = \frac{3}{12} = \frac{1}{4}

• For the sum of two outcomes to be 10, the outcomes must be either (4, 6) or (5, 5) or (6, 4)
• Therefore, P(sum = 10) = P(4, 6) + P(5, 5) + P(6, 4) =  (\frac{1}{12} * \frac{1}{12}) + (\frac{1}{4} * \frac{1}{4}) + (\frac{1}{12} * \frac{1}{12}) = \frac{11}{144}

Hence the correct answer is Option C.

Answer: C

https://e-gmat.com/blogs/wp-content/uploads/2018/07/FT-GC-FB-event-e1532524398949.jpg

HAPPYatHARVARD · Answer

why not probability of 4 = Probability of getting even number * probability of getting 4 out of six numbers.

GMATinsight

GMATinsight · Answer

HAPPYatHARVARD

Once you have included the probability of getting even number then the total choices that you are left with only three numbers {2, 4, 6} hence the following probability can NOT be 'probability of getting 4 out of six numbers', It has to be the probability of 4 out of three even numbers.

PreiteeRanjan · Answer

11/144
Let us assume that the frequency of getting any even number, {2, 4, 6} is x
Then, the frequency of getting any odd number, {1, 3, 5} will be 3x
Implies,
The probability of getting any even number = X/(3X+(3*3X))=1/12
SIMILARLY ODD NUMBER=2/12
TOTAL COMBINATIONS OF GETTING 10 ARE 4+6,5+5,6+4
SO ANSWER WILL BE 11/144

Kinshook · Answer

A biased dice is such that any odd number appears thrice as frequently as any even number.

If the dice is rolled twice, what is the probability that the sum of the numbers thus obtained is 10?

Let the probability that any even number appears be p

3p + 3*3p = 1
The probability that any even number appears (p) = 1/12
The probability that any odd number appears (3p) = 3/12 = 1/4

Throws such that the sum of the numbers thus obtained is 10 = {(4,6), (5,5), (6,4)}

If the dice is rolled twice, the probability that the sum of the numbers thus obtained is 10 = (1/12)*(1/12) + (1/4)*(1/4) + (1/12)*(1/12) = 1/144 + 9/144 + 1/144 = 11/144

IMO C

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

Question of the Week- 20 (A biased dice is such that any odd number..)

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Question of the Week- 20 (A biased dice is such that any odd number..)

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards