Brilliant mega-match

Logic Level 3

You, Calvin, and Pi Han are competing in the Brilliant mega-match! In this event, the winner is the person to solve problems most accurately, the one with the highest ratio of solved problems to problems attempted.* The event runs for five weekdays, and the winner at the end of the week wins it all.

Here are the standings:

Day You Calvin Pi Han
Monday 54% of 100 57% of 700 56% of 500
Tuesday 43% of 200 46% of 300 48% of 300
Wednesday 65% of 100 71% of 200 68% of 400
Thursday 79% of 500 88% of 300 84% of 200
Friday 34% of 100 41% of 500 39% of 100

Calvin performed better than both Pi Han and you in every day, except on Tuesday where Pi Han did better. So who won?

* Of course, if you actually do this, you should give each player a handicap of one incorrect problem so you can't just solve one problem and declare 100% completion. But this is an idealized problem, not a real-world execution of it, so it doesn't matter.

Calvin You Pi Han

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Ivan Koswara by Ivan Koswara , 25, Indonesia

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1 solution

Sharky Kesa
Sep 12, 2015

Though it seems like Calvin and Pi Han have beaten you, you actually have won. This counter-intuitiveness is called the Simpson's Paradox .

During Monday, you answered 54 % × 100 = 54 54\% \times 100 = 54 questions right whereas Calvin answered 57 % × 700 = 399 57\% \times 700 = 399 questions right and Pi Han answered 56 % × 500 = 280 56\% \times 500 = 280 questions right.

During Tuesday, you answered 43 % × 200 = 86 43\% \times 200 = 86 questions right whereas Calvin answered 46 % × 300 = 138 46\% \times 300 = 138 questions right and Pi Han answered 48 % × 300 = 144 48\% \times 300 = 144 questions right.

During Wednesday, you answered 65 % × 100 = 65 65\% \times 100 = 65 questions right whereas Calvin answered 71 % × 200 = 142 71\% \times 200 = 142 questions right and Pi Han answered 68 % × 400 = 272 68\% \times 400 = 272 questions right.

During Thursday, you answered 79 % × 500 = 395 79\% \times 500 = 395 questions right whereas Calvin answered 88 % × 300 = 264 88\% \times 300 = 264 questions right and Pi Han answered 84 % × 200 = 168 84\% \times 200 = 168 questions right.

During Friday, you answered 34 % × 100 = 34 34\% \times 100 = 34 questions right whereas Calvin answered 41 % × 500 = 205 41\% \times 500 = 205 questions right and Pi Han answered 39 % × 100 = 39 39\% \times 100 = 39 questions right.

Furthermore, you answered 1000 questions, Calvin answered 2000 questions and Pi Han answered 1500 questions. We will represent the above information in a table.

You Calvin Pi Han Monday 54 399 280 Tuesday 86 138 144 Wednesday 65 142 272 Thursday 395 264 168 Friday 34 205 39 Total number correct 634 1148 903 Total percentage correct 63.4 % 57.4 % 60.2 % \begin{array} {|c|c|c|c|} \hline & \text{You} & \text{Calvin} & \text{Pi Han} \\ \hline \text{Monday} & 54 & 399 & 280 \\ \hline \text{Tuesday} & 86 & 138 & 144 \\ \hline \text{Wednesday} & 65 & 142 & 272 \\ \hline \text{Thursday} & 395 & 264 & 168 \\ \hline \text{Friday} & 34 & 205 & 39 \\ \hline \text{Total number correct} & 634 & 1148 & 903 \\ \hline \text{Total percentage correct} & 63.4\% & 57.4\% & 60.2\% \\ \hline \end{array}

Clearly, you got a higher percentage of answers correct than Calvin and Pi Han. Therefore, you are the winner.

7 Helpful 1 Interesting 0 Brilliant 0 Confused

Moderator note:

Good explanation of the Simpson's Paradox.

Did in the same way. Simpson's Paradox is fun!

Vinayak Verma - 5 years, 7 months ago

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