1. Distributions Day 1: Review Probability Functions Binomial Distribution Blaise Pascal

2. Warm-up Problem A mechanic fixes 3 bicycles in a day. On each bike, there is a 25% chance that he makes a mistake. Draw a tree diagram to represent this situation. Create a table for the probability distribution of X, the number of mechanical mistakes. Find E(X) Find P(X>0). Given that he makes at least one mistake, find the probability that he makes an odd number of mistakes

3. Warm-up Problem Revisited (Could you do this quickly?) A mechanic fixes 1000 bicycles in a year. On each bike, there is a 25% chance that he makes a mistake. Draw a tree diagram to represent this situation. Create a table for the probability distribution of X, the number of mechanical mistakes. Find E(X) Find P(X>0). Given that he makes at least one mistake, find the probability that he makes an odd number of mistakes

4. Quiz Next Class Basic Probability and Conditional Probability See the most recent assessment Probability Functions Creating a table and/or graph Expected value, or mean of a distribution

5. Today’s Goal Explore a distribution called the “Binomial Distribution” through the traits of algebraic expressions.

6. The Big Problem

7. Problem Set Describe the possible outcomes from tossing: 1 coin 2 coins 3 coins 4 coins

8. Coin Flips, sorted by n(H) Expansions, sorted by power of a 1(1H,0T), 1(0H,1T) 1(2H,0T), 2(1H,1T), 1(0H,2T) 1(3H,0T), 3(2H,1T), 3(1H,2T), 1(0H,3T) 1(4H,0T), 4(3H,1T), 6(2H,2T), 4(1H,3T), 1(0H,4T)

9. Patterns to note, Part 1 Coin Flips Organize the outcomes by “number of heads”. The number of heads decreases from n to 0, while the number of tails increases from 0 to n. Binomial Expansion Organize the terms by degree according to a. The degree (or power) of a decreases from n to 0, while the degree of b increases from 0 to n.

10. Powers of a and b Note that the exponents always have a sum of n (or 5 in the example) Using this reasoning, we can solve part of the Big Problem.

11. The Big Problem Disregarding the coefficient c, what will the x 3 term look like?

12. Starting at ZERO Write your name on the front of the card, and the name of a family pet (or a neighbor’s pet) on the inside (Ignore this…we did it last year)

13. Patterns to note, Part 2 The Coefficients The coefficients exhibit symmetry, with the largest one (or two) in the middle of the distribution How can we get the coefficients?

14. Finding the Coefficients Recursive/Reference method The coefficients may be found on the “n th ” level of Pascal’s triangle, if we call the top level the “0 th ” Explicit Method To find the “r th ” coefficient in an expansion of power “n”, use combinations. You must call the leading coefficient the “0 th ” coefficient.

15. Blaise Pascal (1623-1662) Invented the Pascaline, a mechanical calculator Work on Vacuums and Liquids, Conic Sections, Algebra, Probability, Philosophy and Religion (Pensees) Contemporaries: Fermat, Newton, Leibniz, Cavalieri

16. Pascal’s Triangle [get]

17. Using Combinations Pascal’s triangle has rows and columns. To find the number in the n th row and the r th column (remember that we start counting at zero!), use combinations For example, the work below shows how to find number in the 12 th row, 5 th column.

18. Powers and Coefficients Together

19. The Big Problem

20. Distributions Review Binomial Theorem The General Term

21. After the test C) A six-sided die is rolled 4 times. Determine the probability that a multiple of three will appear exactly 3 times. Hint: the outcomes for each roll should be “Multiple of 3” and “Not a multiple of 3”

22. Binomial Theorem Summary

23. Formula for the rth term in an expansion of the nth power

24. The Big Problem

25. Activity Exercise 4.1.2: 1, 2 (a, e on each) Complete the big problem, if you have not already. Don’t do: 6, 11

26. Distributions Binomial Theorem Review

27. Warm-up 4.1.2: 1c, 6

28. Exercise/Homework 4.1.2 7 11, 12 (you may use Pascal’s Triangle)

29. Analyzing Pascal’s Triangle Powers of 2 Powers of 4 The Fibonacci Sequence The first two Perfect Numbers Perfect Squares Perfect Cubes?