Big numbers are easier to deal with, so we'll start with those, though we all know a ref would never make so many bad calls. Right? We do ... don't we?
If we assume that a 50% of Ref A's calls are bad, and Ref B is equally incompetent, then there is a 75% chance a bad call will be made:
One Crap Ref
A bad (.50) = .50 (.50) +.50
A good (.50) = .50 (.50) +.00
Total 1.00 +.50
50% chance of bad call
Two Crap Refs
A bad (.50), B good (.50) = .25 (.50x.50) +.25
A good (.50), B bad (.50) = .25 (.50x.50) +.25
A bad (.50), B bad (.50) = .25 (.50x.50) +.25
A good (.50), B good (.50) = .25 (.50x.50) +.00
Total 1.00 +.75
75% chance of bad call
Let's say we have particularly experienced and insightful refs and the possibility of making a bad call is only 10%:
One Insightful Ref
A bad (.40) = .10 (.10) +.10
A good (.60) = .90 (.90) +.00
Total 1.00 +.10
10% chance of blowing bad call
Two Insightful Refs
A bad (.10), B good (.90) = .09 (.10x.90) +.09
A good (.90), B bad (.10) = .09 (.90x.10) +.09
A bad (.10), B bad (.10) = .01 (.10x.10) +.01
A good (.90), B good (.90) = .81 (.90x.90) +.00
Total 1.00 +.19
19% chance of bad call
If Esse Baharmast had a twin:
One God-Like Ref
A bad (.01) = .01 (.01) +.01
A good (.99) = .99 (.99) +.00
Total 1.00 +.01
1.00% chance of bad call
Two God-Like Refs
A bad (.01), B good (.99) = .0099 (.01x.99) +.0099
A good (.99), B bad (.01) = .0099 (.99x.01) +.0099
A bad (.01), B bad (.01) = .0001 (.01x.01) +.0001
A good (.99), B good (.99) = .8100 (.99x.99) +.0000
Total 1.0000 +.0199
1.99% chance of bad call
Thus we find that the number of refs does indeed increase the likelihood of bad calls being whistled. Increasing the quality of the ref decreases the likelihood of the bad call being whistled, but not enough to compensate for the additional refs -- no matter how good the refs are. The only way two refs would not increase the likelihood of bad calls is if they are flawless; if they both never bad calls: In other words, if we let parents, players and coaches ref. However, also as shown above, increasing the quality of a single ref does decrease the likelihood of bad calls. Thus, the solution is better refs, not more refs.
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