### Solution to "Another puzzle"

This is my solution to the puzzle from the last post. You probably want to read that other post first, or this discussion won't make any sense at all.

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The puzzle asked for a proof that the rule change does not change the odds of either team winning the game.

It seems like that shouldn't be true: the rule change gives the better team more possessions. It definitely causes many scores to be very, very lopsided in favor of the better team. How can it improve the score, but not change the odds of winning?

In one sentence, the answer is:

Let me show you why. First, I'll give you an explicit "proof", then I'll try to explain why this happens in a sentence or two.

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First, note that under the new rules, if a team passes 100 possessions, it is sure to win. Why? Because, by the rules, the game will be over with the other team having exactly 100 possessions (since we stipulated that the 100th possession is a miss, and when both teams hit 100 or more, the game ends.

At that point, the two teams will have equal numbers of misses (because of equal numbers of "innings"), but the first team will have more possessions. Therefore, it must also have more points.

(So we might as well stop the game as soon as one team hits its 101st possession and declare it the winner!)

Now we show that if the game had gone according to the old rules -- stopping at 100 possessions each, regardless -- that same team would have won.

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The puzzle asked for a proof that the rule change does not change the odds of either team winning the game.

It seems like that shouldn't be true: the rule change gives the better team more possessions. It definitely causes many scores to be very, very lopsided in favor of the better team. How can it improve the score, but not change the odds of winning?

In one sentence, the answer is:

**all those extra points always go to the team who would have won by the old rules anyway**.Let me show you why. First, I'll give you an explicit "proof", then I'll try to explain why this happens in a sentence or two.

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First, note that under the new rules, if a team passes 100 possessions, it is sure to win. Why? Because, by the rules, the game will be over with the other team having exactly 100 possessions (since we stipulated that the 100th possession is a miss, and when both teams hit 100 or more, the game ends.

At that point, the two teams will have equal numbers of misses (because of equal numbers of "innings"), but the first team will have more possessions. Therefore, it must also have more points.

(So we might as well stop the game as soon as one team hits its 101st possession and declare it the winner!)

Now we show that if the game had gone according to the old rules -- stopping at 100 possessions each, regardless -- that same team would have won.

(We're assuming that both teams' first 100 possessions would have been in exactly the same sequence -- although, of course, the teams would have alternated by possession instead of by "inning").

Suppose Team A won under the new rules. Then, both teams had the same number of misses. All Team B's misses came in the first 100 possessions. But at least one of Team A's s misses came *after* 100 possessions. Therefore, Team A had fewer misses than the second team in the first 100 possessions. Therefore, it must have had more points, which means it would have won by the old rules.

Similarly, if a team would have won by the old rules, it would have won by the new rules. Suppose team A won under the old rules. Then, after 100 possessions, it had fewer misses than team B. In that case, team A wouldn't have been stopped at 100 possessions -- it would have kept going somewhere earlier in the game, until it caught up to team B's miss total. Therefore, team A would have had more possessions than team B under the new rules. Therefore, it would have won under the new rules.

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That's basically the solution in a nutshell. The team that hits 100 possessions with the fewest misses will have won, even if it doesn't get any more possessions at all. So any possessions after 100 -- the ones that cause a blowout -- will have occurred after the win was assured. So they can't increase the chances of winning.

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Suppose Team A won under the new rules. Then, both teams had the same number of misses. All Team B's misses came in the first 100 possessions. But at least one of Team A's s misses came *after* 100 possessions. Therefore, Team A had fewer misses than the second team in the first 100 possessions. Therefore, it must have had more points, which means it would have won by the old rules.

Similarly, if a team would have won by the old rules, it would have won by the new rules. Suppose team A won under the old rules. Then, after 100 possessions, it had fewer misses than team B. In that case, team A wouldn't have been stopped at 100 possessions -- it would have kept going somewhere earlier in the game, until it caught up to team B's miss total. Therefore, team A would have had more possessions than team B under the new rules. Therefore, it would have won under the new rules.

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That's basically the solution in a nutshell. The team that hits 100 possessions with the fewest misses will have won, even if it doesn't get any more possessions at all. So any possessions after 100 -- the ones that cause a blowout -- will have occurred after the win was assured. So they can't increase the chances of winning.

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(I haven't mentioned ties yet ... I won't go through the whole explanation, but if there's a tiebreaker required under one set of rules, it would also be required under the other set. And both sets would break the tie the same way.)

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P.S. I always try to explain things too many ways. Let me continue that tradition with one last alternative explanation.

For any sequence of hits and misses, this set of alternative rules would lead to exactly the same score as if you had used the "new" rules:

For any sequence of hits and misses, this set of alternative rules would lead to exactly the same score as if you had used the "new" rules:

Both teams get 100 possessions. The team leading at that time gets to pad its score by taking extra possessions until it has the same number of misses as the trailing team.

The only difference is that under this set of rules, the extra, "over 100" possessions come after the other team's possessions are done, instead of while the other team still has some to go. And it's obvious they don't affect who wins.

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