I read that Roger Federer will be playing Robin Soderling (again) in their upcoming match at Wimbledon.
In their 10 career meetings, Roger Federer has won all the matches against Robin Soderling.
Which means that Federer has won 10 straight matches against Soderling.
So the question is: Who is more likely to win the next match?
Federer or Soderling?
If you interpret it as Federer is much much better than Soderling, then Federer is likely to win the next match.
But if say maybe Soderling has a 1-in-6 chance of winning against Federer on any given day, then doesn’t that mean that it’s high time Soderling wins a match against Federer? (After all, what are the odds of losing 11 times in a row right?)
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woah Jeremy, you just made my day with this question…
I am assuming that someone has to win, which means that Soderling has a 1-in-6 chance and Federer has 5-in-6 chance of winning. It is typically assumed in such probability problems that each match is independent (i.e., results of the past matches have no impact on the outcome of next match). If you can accept the assumption on independence, then the problem is easy. The probability of Federer wining the next match stands at 5/6 while Soderling’s odds remain at 1/6 (i.e., Federer is 5 times more likely to win).
That’s the standard textbook answer that we are taught in probability classes. However, most individuals will probably still feel that a lot of questions are left unanswered. The biggest one is probably the one that you have pointed out; don’t the odds need to “adjusted” to Soderling’s favor so that we will eventually see 1-in-6 in the long run?
Shucks… I realize that this can really go on for quite long. And there is still the assumption of independence and the assumption of a stationary probability distribution. Hmmm… now I really want to write about these. But let me give a short answer to the question I have posed.
Contrary to our intuition, Soderling’s odds do not require any adjustment for there to be a Soderling to have 1/6 number of wins in the long run (which I suspect is the reason why people intuitively believe that there is overwhelming odds that the next match is Soderling’s) . This is proven in the Law of Large Numbers. The proof can be found on Wikipedia. I suggest that you check out the one that uses Chebyshev’s inequality. The second proof assumes that you know what are characteristic functions. Haha… but unless you are really into the field of probability, that assumption is violated.
http://en.wikipedia.org/wiki/Proof_of_the_law_of_large_numbers
Whoa Chin Hon!!! Cheem analysis!
Hmmm, yup I get the part about “independence”. I guess if we assume independence then it’s quite easy to understand.
Your last paragraph is quite interesting. I never knew that before.
Thanks for shedding some light!
Wa lau. I cannot match the PhD’s answer lah. Mine too simplified. Only Ah-Level standard… heh…
haha… and i am only an ah beng/ah seng…
Oh, just for the record…Federer won the match against Soderling. =)