The Magic Number

Lose Remerswaal

Lose Remerswaal

Experiencing Furry Panic
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Nov 15, 2002
50,362
Dicker and Dicker of Beverly Hills
It's the middle of August and yesterday someone mentioned the Sox' Magic Number and I was like, WOW! We can talk magic numbers and it's so early in the season! Let's keep the other thread as an homage to how great this team is.

So why not? These are fun times, so have fun here, posting your favorite version of the Sox' current magic numbers!

Boston RedSox – From the green walled confines of Fenway Park, we tender this baseball team for your enjoyment as a tribute to your great fandom. It comes from the Back Bay to you.

 
EdRalphRomero

EdRalphRomero

wooderson
SoSH Member
Oct 3, 2007
4,446
deep in the hole
I always like to track the number of remaining outcomes a team has to work with to get to the Magic Number. For today, that number of remaining outcomes (Red Sox games outstanding plus MFY games outstanding) is 84.
 
loshjott

loshjott

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Dec 30, 2004
14,581
Silver Spring, MD
Lose Remerswaal said:
It's the middle of August and yesterday someone mentioned the Sox' Magic Number and I was like, WOW! We can talk magic numbers and it's so early in the season! Let's keep the other thread as an homage to how great this team is.

So why not? These are fun times, so have fun here, posting your favorite version of the Sox' current magic numbers!

Boston RedSox – From the green walled confines of Fenway Park, we tender this baseball team for your enjoyment as a tribute to your great fandom. It comes from the Back Bay to you.

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As someone who drank way too much of that swill in college, thanks for the reference.
 
Lose Remerswaal

Lose Remerswaal

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Dicker and Dicker of Beverly Hills
EdRalphRomero said:
I always like to track the number of remaining outcomes a team has to work with to get to the Magic Number. For today, that number of remaining outcomes (Red Sox games outstanding plus MFY games outstanding) is 84.
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I like that! So in a 50-50 world, the Sox chances of success are 51 out of 84?

loshjott said:
As someone who drank way too much of that swill in college, thanks for the reference.
Click to expand...
Same here. Oh so same here.
 
EdRalphRomero

EdRalphRomero

wooderson
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Oct 3, 2007
4,446
deep in the hole
Thanks @Savin Hillbilly That is some good stuff. I guess Lose's math was directionally correct.

For me, I feel good saying we just need 33 out of 84 outcomes to bounce our way. (Let me check to see if Eddie Gaedel ever spent time with the Sox wearing 33/84 for a number.)
 
Savin Hillbilly

Savin Hillbilly

loves the secret sauce
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Jul 10, 2007
18,783
The wrong side of the bridge....
Red Sox Physicist said:
You are missing binomial weighting. There's only one combination where the Red Sox to win all 41 remaining games, but there are C(41,k) = 41!/(k!(41-k)!) combinations where they win k games.
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D'oh! Of course. I knew something was fishy there, but couldn't put my finger on it. Well, there's twenty minutes of my life I'll never get back. Apologies for the waste of bandwidth.
 
Red Sox Physicist

Red Sox Physicist

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Jul 15, 2005
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Natick, MA
EdRalphRomero said:
So...could you be persuaded to do the binomial math for us?
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Sure. It's complicated by the 6 mutual games left between the Yankees and Red Sox. I'm ignoring the fact that Toronto and Tampa aren't technically eliminated yet either, and assuming each remaining game is a coin flip. This is ignoring the tie case as well.
Here's a MATLAB script that calculates it:

mutual = sym(6); % 6 mutual Red Sox/Yankee games
totalgames = sym(78); % 78 total games left between the Red Sox and Yankees
nonmutual = totalgames - mutual; %Number of nonmutual games left
probweight = sym(1/2)^totalgames; % Assume a 50% probability of each game outcome
clinchnum = sym(33); % Current clinch number is 33 total Red Sox wins + Yankees losses

p = sym(0);
% Separate out non mutual games and Yankees/Red Sox games
% rwin is the number of Red Sox wins in mutual Red Sox/Yankees games.
% The clinch number drops by 2 for every Red Sox win in mutual games
for rwin = sym(0):mutual
p = p + sum(arrayfun(@(x)nchoosek(nonmutual, sym(x)),(clinchnum-sym(2)*rwin):nonmutual))*nchoosek(mutual,rwin)*probweight;
end

eval(p)*100
97.4%
 
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Average Reds

Average Reds

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Sep 24, 2007
35,040
Southwestern CT
Red Sox Physicist said:
Sure. It's complicated by the 6 mutual games left between the Yankees and Red Sox. I'm ignoring the fact that Toronto and Tampa aren't technically eliminated yet either, and assuming each remaining game is a coin flip. This is ignoring the tie case as well.
Here's a MATLAB script that calculates it:

mutual = sym(6); % 6 mutual Red Sox/Yankee games
totalgames = sym(78); % 78 total games left between the Red Sox and Yankees
nonmutual = totalgames - mutual; %Number of nonmutual games left
probweight = sym(1/2)^totalgames; % Assume a 50% probability of each game outcome
clinchnum = sym(33); % Current clinch number is 33 total Red Sox wins + Yankees losses

p = sym(0);
% Separate out non mutual games and Yankees/Red Sox games
% rwin is the number of Red Sox wins in mutual Red Sox/Yankees games.
% The clinch number drops by 2 for every Red Sox win in mutual games
for rwin = sym(0):mutual
p = p + sum(arrayfun(@(x)nchoosek(nonmutual, sym(x)),(clinchnum-sym(2)*rwin):nonmutual))*nchoosek(mutual,rwin)*probweight;
end

eval(p)*100
97.4%
Click to expand...

Check out the big(ger) brain on Red Sox Physicist!
 
Reverend

Reverend

for king and country
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Average Reds said:
Check out the big(ger) brain on Red Sox Physicist!
Click to expand...
The last time I saw this brought up on SoSH, it was only for a seven game series (it's a fairly standard Statistics 101 type question), and I watched two SoSHers race to produce the answer using different methods.

If I recall correctly, @ToeKneeArmAss basically made some kind of mechanical Turk out of spreadsheets in Excel to reproduce what the mathematical formula did. I wonder if I still have it saved somewhere because it was basically SABR art. I mean, it was to art what brutalism is to architecture, but it definitely had an aesthetic.
 
The Raccoon

The Raccoon

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Jul 24, 2018
898
Germany
I did some poor mans excel work earlier to check, if I get the same probabilities as Red Sox Physicist in post #18 (which I did) and to see what impact the 6 remaining games between the Sox and the Yankees will have on the outcome of the AL East.
I also assumed that the Blue Jays and Rays will not win the division and that all 75 remaining Yankees and Red Sox games will be coin flips.

Here are my results:
If the Red Sox sweep the Yankees in the 6 remaining games, they will win the division with a likelihood of 99.996%
Red Sox win 5 games: 99.968%
... 4 games: 99.823%
... 3 games: 99.228%
... 2 games: 97.336%
... 1 game: 92.598%
... 0 games: 83.222%

Obviously a sweep for either side is far more unlikely (1.6% each) than a 3-3 record (31.3%) if we assume 50-50 probabilities for each individual game, so the total weighted probability for winning the division sits at 98.134% right now.

Edit: All probabilites are calculated based on the Red Sox winning at least one more game than the MFY. Therefore the probabilities could even be slightly higher, if we include the TIE scenario and a game 163, that could be won by the RS...
 
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rajendra82

rajendra82

elimination day disfunction
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Jul 16, 2005
4,925
Atlanta, GA
I wrote my own simulation spreadsheet below that makes each game a coin toss. Ran it multiple (i.e. 208) times. 203 times the Red Sox won the division, 4 times the MFYs did, and when I got the 1978 like result, I stopped pressing F9. Crudely estimated outright division win probability 97.5%.

https://www.dropbox.com/s/2fc7m2u2oe6wjf6/2018.xlsx?dl=0
 

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