First of all, I’m going to play for time. Below, I’m going to give away the answer to the Bob’s Boast problem (in my previous blog post). So right now, on your screen, you aren’t seeing the answer. Instead, here’s a link to a lovely and pretty darn cheap fantasy novel I wrote years ago. It literally has a 5-star rating on Amazon, and I don’t actually know the person personally. So, go look, and if you have 99 centaroos clanking around in your virtual pocket, buy Princess of Ghosts. You’ll like it a lot: that’s a statistical fact.
Remember Bob’s boastful boasts? Bob has a higher batting average for the first half of the season than Sally does. And Bob also has a higher batting average than Sally the second half of the season. Yet Sally claims her overall batting average for the whole season is higher than Bob’s. And they actually had the same number of at bats. Can this be?
Suppose that in the first half of the season, you have these numbers:
- Sally had 50 at bats, and hit .100
- Bob had 200 at bats, and hit .200
And in the second half, you have:
- Sally had 250 at bats, and hit .400
- Bob had 100 at bats, and hit .500
That means that:
- In the first half, Sally had 10 hits, and in the second half, she had 100 hits
- In the second half, Bob had 40 hits, and in the second half, he had 50 hits
So overall, they both had 300 at bats. Then the batting averages are:
- Sally had 105 hits / 300 at bats, for a batting average of 3.50
- Bob had 90 hits / 300 at bats, for a BA of .300
Sorry, Bob! You did best in the part of the season when you played least. But her playing time was much less when both of you were doing badly. Since she had the most hits at the best time (and the least hits when she was doing the worst), Sally beat you. Again.
(There are surely an infinite number of solutions: numbers of hits and at bats in different parts of the season which allow for the overall winner to not be the winner of any part of the season.)
What do you think?