Wheel of ProbabilityBernard Brooksyoutube link

Wheel of Probability• Probability and number types• http://mathmidway.org/Training/pdf/numberlineTourGuide.pdf • Number Devil book • How many primes from 1-100: • Sieve of Eratosthenes

• Binomial Tree and coin flipping

Activities• Sieve of Eratosthenes• Fibonacci tree• Binomial tree• Betting: If I am selling a ticket that pays $10 if you spin an even

number and nothing if you get odd, how much is that ticket worth? • Hint: less than $5

Sieve of Eratosthenes

• Why isn’t 1 a prime number?

Exhibit Prompts• Probability of getting• Even number• Number less than 30• Triangle number (1,3,6,10,..)• Prime number

• Uniform distribution of one spin but take the average of spins and it approaches a Gaussian

• Betting on the wheel: What does a $1 wager that pays off if you spin a 29 have to pay to be a worthwhile bet?

Number Names• Prime numbers• Sieve of Eurastonese

• Fibonacci numbers• Trees and pineapples

• Squares• Triangle numbers• Sum of 1 to n

• Perfect numbers

Fibonacci Tree• What is the probability of getting a Fibonacci number?• Tree rule: a branch needs to mature before it can birth a new

off shoot. Start with a new shoot.

• Pine cones• Pineapples• Flowers

Binomial Tree• Start the children all in a line (at 0 on a number line)• Odd: step forward, even backward • After 10 steps graph where everyone is: Gaussian curve• Prompts: what’s the farthest you can be? What is the chance that

happens?• What about the rule: 30 step back, > 30 step forward?• Can be done with coin flipping (10 coins at once)• Mean value or expected value of np• Variance as expected value of (x-np)2

Excel Plot of Binomial Tree

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Social Network• Binomial Random Network• With 10 people how many connections are possible? (it’s a

triangle number)• With p=1/2, how many connections are expected?• Draw the degree distribution and see it’s Gaussian

• Preferential Attachment Network• Start with kite graph• Add people with 2 edges • Rich get richer

Simple Rules • Coin flipping• Stock prices• Yes/no questions• Binomial probability• Random social networks• Pascal’s Triangle• Binomial Coefficients

Questions?

Wheel of Probability Prompts• What is the probability of getting• Even number• Number less than 30• Triangle number (1,3,6,10,..)• Prime number

• If we all spin the wheel once and graph the distribution of our results we get a uniform distribution. • If we spin 2 times and average them, 3 times and average, etc. it

approaches a Gaussian• What does a $1 wager that pays off if you spin a 29 have to pay

to be a worthwhile bet? • We see Fibonacci numbers in nature more often than chance

would dictate; why?