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THEORETICAL MODEL 1A
For my research, I will use a Bayesian approach, which provides the flexibility to keep the prior odds separate from the conditional likelihood. This will make it possible to measure how much weight the coaches are giving new information compared to the original information when making decisions, in turn allowing me to directly test for the representativeness heuristic.
THEORETICAL MODEL 1B
This is Bayes' rule in odds form.
On the left hand side of the equation we have what is called the posterior, which in this circumstance is the in-game odds of team C winning the game given the fourth down decision that the coach made.
THEORETICAL MODEL 1C
The first term on the right hand side is called the prior, which is the odds of team C winning the game given the game-state, while the second term is called the conditional likelihood, which is the inverse conditional odds of team C making that decision given that they won the game.
In other words these two components are the original information (the coin being a 50|50 chance of landing on heads) and the new information (the fact that the first five tosses happened to land on heads).
THEORETICAL MODEL 1D
It's the second term on the right-hand side (the conditional likelihood) that can be hard to wrap your mind around, because it's rather counter-intuitive. You aren't looking at the odds of winning the game, you're looking at the odds of that decision being made GIVEN that they actually win the game.
[pause here – let it sink in]
So you take teams that have been in similar situations in other games and found how likely (or unlikely) they were to make that decision given that they did or didn't win the game.