Worse Off Despite Leading

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One of my Random Writings.

2007/07/03 Worse off, despite leading...

Suppose a tennis player has a 95% chance of winning any given point on their serve.
For reference, with 95% chance on each point, you have 99.991% chance of winning the game from love all, but "only" a 99.986% chance from 40:30. That's 0.005% less.

Miniscule, true, but you'd think it shouldn't happen at all.

Some people claim that it's not unreasonable for this to be true. Fair enough. Tell me then. I've talked about the case where P=P(server wins point)=95%. For what values of P does this "paradoxical" result occur?

Amazingly, they have less chance of winning the game from 40:30 than they have from Love All, even though 40:30 means they are in front!

This is the sort of puzzle that intrigues me. You can't believe it, then you do the sums and it turns out to be true! How amazing is that! Surely this is the sort of thing to inspire kids to study maths, to find out why it happens.

Then again, maybe not. In my experience, some kids will be intrigued, but others will have a strong adverse reaction. "Surely," they say, "This is nonsense, and only goes to show how irrelevant maths is to everyday life. It can't be true, I won't believe it, and maths is stupid for claiming things like this."

Hmm. Perhaps some things aren't good advertising, even when they are true. The point is, though, that it is exactly this sort of result that shows we need mathematics. When your intuition leads you astray, maths can come to the rescue. It's maths that can confirm or refute apparently bizarre claims, and save you from wasting time, wasting money, and in extreme cases, can save lives.

Extracted from Colins Blog 2007.



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