Informal Problem Solving

One thing I enjoy reading is when problems are solved by common sense rather then brute force. One example is from the Sabermetric Research Blog in a post titled 'Why 10 runs equals 1 win'. The point to realise is that an extra run are added at random. It may come up when down one in the bottom of the ninth but, equally, it may come when up six in the ninth.

The only time the extra run matters is when a team lost by one or went to extra innings.

When a team lost by one the extra run brings them to extra innings where they have a 50% chance of winning. Since 22.5% of games are decided by one run, 11.3% of the time the extra run goes to the team that lost and that converts to  an a win 5.6 of the time - or 0.56 wins.

9% of games go to extra innings. If you add a run to one of those teams then they win the game - but they would have won half of them anyway. So the extra run matters 4.5% of the time and that converts to 0.45 wins.

0.56 wins + 0.45wins = 1.01 wins (Remember your denominonsense).

A more recent example that I have seen of good critical thinking is from the book 'Fifty Challenging Problems in Probability with Solutions' by Frederick Mosteller. In question 6, Chuck-a-Luck, he poses the following question:

How would you approach it?