Fraction Brainteaser

Source:
Sent to me by Gaurav Sinha

 Problem:
Siddhant writes a Maths test and correctly answers 5 out of 6 Arithmetic questions and 20 out of 28 Geometry questions. In total, Siddhant scores 25 out of 34. 

 Vaibhav writes another Maths test and correctly answers 20 out of 25 Arithmetic questions and 6 out of 9 Geometry questions. in total, Vaibhav scores 26 out of 34.

 Note that
a) Vaibhav scores more than Siddhant
b) Siddhant score better than Vaibhav in both individual topics - 5/6 > 20/25 and 20/28 > 6/9

 How is it possible? 

Solution:

Highlight the part between the * symbols for the answer.
* This is Simpson's Paradox: each topic on its own shows one trend, but the combined score shows the opposite.

Arithmetic is the easier topic - both students score a higher fraction there (Siddhant 5/6 = 83.3%, Vaibhav 20/25 = 80%) than in Geometry (Siddhant 20/28 = 71.4%, Vaibhav 6/9 = 66.7%). But the two tests mixed the topics very differently. Vaibhav's 34 questions were mostly Arithmetic (25 easy, 9 hard), while Siddhant's were mostly Geometry (6 easy, 28 hard). So Vaibhav's overall score is dominated by his easy-topic rate and Siddhant's by his hard-topic rate.

Combining: Siddhant (5+20)/(6+28) = 25/34 = 73.5%, Vaibhav (20+6)/(25+9) = 26/34 = 76.5%. Siddhant wins each topic but loses the total - the very different topic weights, not the per-topic rates, decide the aggregate. (solution posted by AI) *

Comments

  1. mbj11112:55 PM, December 15, 2017

    Score = accuracy * total score
    So, for Siddhant higher accuracy (5/6) has lower scoring (6) where as Vaibhav has higher score in more accurate tests(arithmetic)

    ReplyDelete
  2. GoKu1:41 PM, December 15, 2017

    This is Simpson's paradox. Siddhant is better than Vaibhav in each individual subject, but Siddhant's total score is lesser because he attempts more geometry questions, which are harder than arithmetic questions.

    See https://en.wikipedia.org/wiki/Simpson%27s_paradox for a more detailed explanation.

    ReplyDelete
  3. Sumit Bhagwani2:30 PM, December 15, 2017

    Score is just the sum of the correct answers. It has nothing to do with fractions.

    ReplyDelete
  4. sumanth10:11 PM, January 02, 2018

    each question of Arithmetic =x, Geometry =y (say)
    6x+28y=34, 25x+9y=34
    => x=y=1
    Therefore, each has same weightage of 1 mark.
    Even though, 5/6 > 20/25 and 20/28 > 6/9 Vaibhav answered more (20+6=26) questions than Siddhant(5+20=25)

    ReplyDelete
  5. random1237:41 PM, January 23, 2018

    Let C denote solving problem correctly. Let S denote solving by Siddhant and A denote Arithmetic problem.
    Then according to problem P(C | S, A) > P(C | ~S, A) and P(C | S, ~A) > P(C | ~S, ~A) but P(C | S) < P(C | ~S).
    This problem is very prominent in statistics. Here the variable A is called the confounding variable, it influences both the independent (S) and dependent variable(C). More can be understood from wiki link on "Confounding".

    ReplyDelete
  6. Unknown10:13 AM, January 24, 2018

    Because a/b+c/d != (a+c)/(b+d)

    ReplyDelete
  7. Mukul Bansal10:20 AM, August 07, 2018

    Although, in both concepts, Siddhant is better than vaibhav. But see that, both are more good in arithmetic than the geometry. Means if, more arithmetic questions are put, the percentage of total score will go up. Thus for vaibhav due to more questions from arithmetic, he has higher total score.

    ReplyDelete

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