1. Mathematics in Movies

2. 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
_________ = → actual win percentage: 0.636 ​

3. 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

5. 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

6. 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

8. 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

9. 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

11. 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.

12. 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...