Mathematics in Movies

Moneyball Pythagorean Expectation runs scored2 Discovered by statistician Bill James Estimates win percentage in terms of runs scored and allowed runs scored2 _________________________ runs scored2 + runs allowed2 Ex: Oakland A’s 2002 season → 800 runs, 654 runs allowed 8002 _________ = 0.599 → actual win percentage: 0.636 8002 + 6542 ​

Sabermetrics Used to identify and hire players based on statistics → “high on-base percentages, good baserunning skills, and solid defensive abilities” (Rykoff) Ex: OPS (On-Base Plus Slugging): on-base percentage + slugging percentage → looks at how well a player gets on base and hits for power wOBA (Weighted On-Base Average): batting average that factors in and differentiates the types of hits that players make

Rain Man Counting Cards When following basic strategy, the house has an edge of .2% to .6% In reality the house edge is constantly changing because cards are being revealed Counting cards uses conditional probability to somewhat keep track of that edge When counting cards you adjust your bets based on what’s called “the count” The count is a representation of how likely it is for a 10, J, Q, K, or A to be drawn

Rain Man How it works: 2-6 are valued at +1 7-9 are valued at 0 10-A are valued at -1 Counting cards turns blackjack into a long term game rather than just playing each hand at a time i.e. You lose 4 hands in a row, but the count is at +10, you can bet high and most likely come out with a net profit By simple addition and subtraction, you keep track of the count The higher the count, the more likely a 10-A is to be put down

A Beautiful Mind Game Theory: Zero Sum Noncooperative games: In mathematics, a game involves two or more players Players earn rewards or penalties based on the actions of all participants One player’s gain is another player’s loss Noncooperative games: Players can change their strategy to improve their own outcome without affecting the other players

A Beautiful Mind Example of Noncooperative Games: Prisoner’s Dilemma Two criminals are detained in separate cells Going to be charged with a lesser crime, spending 1 year in prison, unless they testify If they testify against the other they go free, while their partner gets 3 years If both betray each other than they both serve 2 years Nash Equilibrium John Nash discovered that if both players do what’s best for both themselves and the team then the outcome will be the most beneficial

Mean Girls Direct substitution gets 0/0 L'hopital's Rule: Example: limx->0 x/x^2+x Differentiate top and bottom to get 1/2x+1 Plug in 0 to get 1 as the limit.

Mean Girls Applying L'hopital's Rule ((1/(1-x))-cos(x))/sin2x ln(1-x) becomes -1/1-x -sin(x) becomes -cos(x) 1-cos^2(x) can be written as sin^2x, which becomes sin(2x) ((1/(1-x))-cos(x))/sin2x Plug in 0’s to get ((-1/(1-0))-cos(0))/sin2(0) or ((-1)-1)/sin(0) sin(0) is 0, and dividing by a 0 is undefined, therefore...