Early on Saturday, we took to Twitter and asked people for suggestions for quick questions we could ask the pros at this Grand Prix. The yield wasn't particularly big, but a certain someone came up with a question that really intrigued us.
@magicprotour If each individual player in your team wins 60% of their matches, what is the probability that your team wins the round?— Frank Karsten (@karsten_frank) May 16, 2015
Here is what our pros came up with ...
Ondřej Stráský, two time Pro Tour Top 8 competitor: [Without knowing who posted the question in the first place.] "I would ask Frank Karsten!"
Reid Duke, No. 21 in the Top 25 Rankings: [Took some time to run some calculations in his head ...] "64.8% of the time!" [Which is a hundred percent correct.]
Martin Jůza, 22 Grand Prix Top 8s: [Making a wild guess ...] "83.6%!"
Eduardo Sajgalik, Top 8 at Grand Prix Milan 2015: [Pulled out his notepad, started crunching the numbers, and came up with ...] "64.8%!"
Valentin Mackl, five Grand Prix Top 8s: "I don't care. It's over 60%!"
Wenzel Krautmann, champion of Grand Prix Warsaw 2013: "I'm a lawyer, not a mathematician! I don't know this stuff."
Now you might be interested in a somewhat longer answer, like how to work out the result yourself. We can help you there. When doing probability calculations, there are three simple—well, simplish—rules one needs to keep in mind:
- "And" means multiplication. The probability that both thing A and thing B will happen, if they are independent, is the probability of A times the probability of B.
- "Or" means addition. The probability that A or B will happen, if they are exclusive, is the probability of A plus the probability B.
- Very important: You need to account for all of the possible combinations that lead to the outcome in question.
Now onto the problem at hand! The answer could be given in the following form: It's the probability that your team wins the first and the second and the third match, or wins the first and the second match and loses the third, or wins the first and the third match and loses the second, or wins the second and the third match and loses the first.
Translating that into a formula by simply turning every "and" into a multiplication and exchanging every "or" for a plus sign, we get:
0.6 × 0.6 × 0.6 + 0.6 × 0.6 × 0.4 + 0.6 × 0.6 × 0.4 + 0.6 × 0.6 × 0.4 = 0.648