| Subject: Why Going For Two Points When Down 8 Works, Mathematically |
Author:
An Observer
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Date Posted: 14:32:11 09/23/24 Mon
In reply to:
Northbounder
's message, "Re: Numbers, Numbers, Numbers. I Love Numbers." on 19:02:42 09/22/24 Sun
Northbounder, I called a couple of my friends who are mathematically inclined and discussed your post with them, guys who are in the numbers business. You have just ruined the last 24 hours for us because we have been wrestling with not just HOW your two-pointer strategy works, but *WHY*.
A few minutes ago, I was standing at a urinal taking care of some non-numbers business when suddenly the light bulb finally turned on in my head.
It seems that, mathematically, by going for two points down by eight in the fourth quarter, the scoring team manufactures out of thin air the extra 12.5% probability of winning you say results from that choice. How can that possibly be? How can 12.5% probability of winning simply materialize from choosing from nothing but 50% outcomes (making a two-pointer and winning in overtime?
It's taken me a full day to figure it out.
Let's call the two strategies KICK-KICK and TRY-TRY.
The expected value of KICK-KICK is 2.
Because the probability of converting a two-pointer is 50%, the expected value of TRY-TRY is also 2.
So why is TRY-TRY the better strategy?
Because part of the value of TRY-TRY comes from scoring 4 points with MAKE-MAKE. The expected value of 2 comes from 0.25% probability of each of these outcomes:
FAIL-FAIL = 0 prob-adjusted value of this outcome = 0.0
FAIL-MAKE = 2 prob-adjusted value of this outcome = 0.5
MAKE-FAIL = 2 prob-adjusted value of this outcome = 0.5
MAKE-MAKE = 4 prob-adjusted value of this outcome = 1.0
Expected value of TRY-TRY = 8/4 = 2.0
But as discussed, if the team MAKES the first try, it will kick after the second touchdown. There will never be a MAKE-MAKE outcome.
So the scoring team sacrifices the possibility of scoring 4 and improves the possibility that they will score 3.
FAIL-FAIL = 0 prob-adjusted value of this outcome = 0.0
FAIL-MAKE = 2 prob-adjusted value of this outcome = 0.5
MAKE-KICK = 3 prob-adjusted value of this outcome = 1.5
Expected value of TRY-(TRY or KICK) = 2.0
But scoring 3 wins you the game outright, whereas scoring 2 only gives you a 50% probability of winning in overtime.
The scoring team is shifting expected value from MAKE-MAKE and moving it to MAKE-KICK.
That's where the extra probability of winning comes from.
The scoring team is giving up the useless value of [the probability of winning by two] and increasinging [the probability of winning by one or in overtime].
It's taken me a full day to realize this and explain it to my math friends. Gotta use the urinal more often.
Thank you, Northbounder, for a fascinating 24 hours in our lives.
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