Word Problems = $1.10 1 Ball + (1 Ball + $1.00) A baseball and bat together cost $1.10. The bat costs $1.00 more than the ball. What does the ball cost?
Word Problems = $1.10 1 Ball + (1 Ball + $1.00) A baseball and bat together cost $1.10. The bat costs $1.00 more than the ball. What does the ball cost? 1 Ball + 1 Ball + $1.00 = $1.10
Word Problems = $1.10 1 Ball + (1 Ball + $1.00) A baseball and bat together cost $1.10. The bat costs $1.00 more than the ball. What does the ball cost? 1 Ball + 1 Ball + $1.00 = $1.10 2 Balls + $1.00 = $1.10
Word Problems = $1.10 1 Ball + (1 Ball + $1.00) A baseball and bat together cost $1.10. The bat costs $1.00 more than the ball. What does the ball cost? 1 Ball + 1 Ball + $1.00 = $1.10 2 Balls + $1.00 = $1.10 2 Balls = 10¢
Word Problems = $1.10 1 Ball + (1 Ball + $1.00) A baseball and bat together cost $1.10. The bat costs $1.00 more than the ball. What does the ball cost? 1 Ball + 1 Ball + $1.00 = $1.10 2 Balls + $1.00 = $1.10 2 Balls = 10¢ 1 Ball = 5¢
James Edward Gray II • I’m a programmer • We use at least a little math in our jobs • I’m an atheist • Good atheists use a little math too • I’m not a math whiz
Branches of Math • Combinatorics: dealing with combinations of objects • Estimation: rough calculations of value, quantity, or extent • Probability: the extent to which an event is likely to occur
Math Quiz How many people would you need to gather to be absolutely certain (a 100% chance) that two people in the group share a birthday (month and day only)?
Math Quiz: Round 2 How many people do we need to gather if we are willing to accept half the odds (a 50% chance) that two people in the group share a birthday?
Birthday Combinatorics • Ignoring Leap Year to keep this simple… • The possible combinations of five different birthdays: 365 × 365 × 365 × 365 × 365 (or 3655) • The ways you can select five birthdays without repeats: 365 × 364 × 363 × 362 × 361
Birthday Probability • The ratio (fraction) of events we’re interested in to the total possible events • Chance five people don’t share a birthday: (365 × 364 × 363 × 362 × 361) / (365 × 365 × 365 × 365 × 365) • Subtract from one to get the chance of the opposite occurrence (a birthday is shared)
Math Quiz: Round 3 You visit a friend, knowing she has two children but not the gender of the children. A young girl answers the door, so that’s one child. What are the odds that the other child is a boy?
Math Quiz: Round 3 You visit a friend, knowing she has two children but not the gender of the children. A young girl answers the door, so that’s one child. What are the odds that the other child is a boy? ⅔
Are They on Our Side? Image from http://sharetv.org/shows/the_big_bang_theory #1: http://tvbythenumbers.zap2it.com/2012/04/17/syndicated-tv-ratings-big-bang-theory-on-top-again-dr-phil-leads-talk-shows/129421/
Danica McKellar is on Our Side • Actress (Wendy on The Wonder Years) • Sex symbol • Math degree (with honors) • Published author • Chayes-McKellar-Winn theorem Image from http://allinone1-z.blogspot.com/2010/12/danica-mckellar-hotdanica-mckellar.html
U.S. Math Literacy • The U.S. is below a 34 country average • 17 of those 34 countries score better than we do while only 5 score below us (11 are roughly even with us) • Ranking math proficiency from level 1 to level 6, only 27% of U.S. students score 4 or higher 2009 statistics for 15-year-olds from http://nces.ed.gov/surveys/pisa/pisa2009highlights_3.asp
A Biblical Flood • “…all the high mountains under the entire heavens were covered.”—Genesis 7:19 NIV • 10,000 to 20,000 feet of water • “And rain fell on the earth forty days and forty nights.”—Genesis 7:12 NIV • 960 hours of rainfall • That’s water rising about 15 feet per hour
"The State Department is infested with communists. I have here in my hand a list of 205— a list of names that were made known to the Secretary of State as being members of the Communist Party and who nevertheless are still working and shaping policy in the State Department."—Joe McCarthy, U.S. Senator
Math Quiz Judy is thirty-three, unmarried, and quite assertive. A magna cum laude graduate, she majored in political science in college and was deeply involved in campus social affairs, especially in anti-discrimination and anti-nuclear issues. Which statement is more probable? (a) Judy works as a bank teller. (b) Judy works as a bank teller and is active in the feminist movement.
Occam’s Razor (a) Judy works as a bank teller. (b) Judy works as a bank teller and is active in the feminist movement. Bank Teller Odds * Feminist Odds < Bank Teller Odds
Polling • Any kind of polling is subject to statistical errors • This is where our margin of error comes from • Even that can (rarely) be off • You also have to worry about systemic errors
Gerrymandering • Nick had to move to even try to get elected • That’s because politicians use packing and cracking to distribute votes • Nick’s district looks like a dinosaur • The Voting Rights Act made this illegal, but only when racial (political is fine)
The Census • The constitution requires an “actual enumeration” of the population every 10 years • This process costs billions • It contains known systemic errors
Is Sampling Legal? • In the past, the U.S. Census used both approaches • This has been fought out in the Supreme Court: • Sampling was declared unconstitutional • Imputation is still allowed
It Takes More Than Just Math • Two gamblers wager $100 on coin flips • The first to win six flips takes the pot • Their game is interrupted when the score is five to three • How do we fairly divide the pot?
Payout Arguments • Pay $100 to the gambler with five wins since he’s leading and the prize is all or nothing • Payout ⅝ and ⅜ of the pot for their respective wins • Pay ⅞ and ⅛ for the odds each man will win (½ × ½ × ½ = ⅛ and 1 - ⅛ = ⅞)
Math Quiz You need to test the blood of 100 patients for a disease. On average one in a 100 will have the disease. Can you find those that have the disease by doing less than 100 tests? (Givens: you have enough of each patient’s blood to test them several times and you can test multiple patients at once by combining samples.)
A Good Rule of Thumb • A lot of people choose to look up a new word when they first encounter it • It’s an equally good idea to look up new math when you encounter it • http://www.wolframalpha.com/ • Ask a math geek
The Formulas P = L(i(1 + i)m) / ((1 + i)m - 1) B = L(1 + i)m - (1 + i)p) / ((1 + i)m - 1) P = Monthly payment L = Loan amout i = Monthly interest (6% loan: 0.06 / 12 = 0.005) m = Length of the loan in months B = Balance remaining p = Payments made
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