Minds on! CP Canoe Club Duck Derby Prizes Maximum of 2 000 tickets sold $5/ticket Prizes Group of 12 VIP tickets to Cirque du Soleil ($2000) 32GB BlackBerry Playbook ($600) $250 cash Should Mr. Lieff buy a ticket?

Expected Value Calculation For 2 000 tickets E(X) =(1/2000)(2000) + (1/2000)(600) + (1/2000)(250) = 1.425 or $1.43 return on a $5 ticket Suppose Mr. Lieff receives a hot tip that ticket sales have stalled at 1 500 E(X) =1/1500 (2000) + 1/1500(600)+1/1500(250) = 1.9 or $1.90 return on a $5 ticket

Real-world probability: Basket Draw In a basket draw, tickets cost $1 There are a number of prize packs – to enter you put any number of tickets in a basket One winner is drawn from each basket What are some possible strategies?

Basket Draw – what I won $150 Miken softball glove (sold) 1 night at Qualify Suites London (gifted) $55 at Fox and Firkin, Toronto (gifted) $50 + 15% off + 2 free apps Montana’s London (used) $50 National Sports (used) $25 TBS (used) 2 bottles of wine, Bailey’s, rum, 4 Mott’s Caesar, 2 VEX Duck Dynasty hat (gifted) Skittles, microwave popcorn, cherry cordials (gifted/eaten) Total value: $600

5.1 Probability Distributions and Expected Value Chapter 5 – Probability Distributions and Predictions Questions: Games Fair? Learning Goals: Create probability distributions and calculate Expected Value MSIP/Home Learning: p. 277 #1-5, 9, 12, 13

7-day weather forecast What is POP? Probability of Precipitation What does it mean?

What is a probability distribution? A set of data consisting of all possible outcomes and the associated probabilities e.g., weather forecast for March 7-12, 2016

Probability Distributions of a Discrete Random Variable a discrete random variable, X, is a variable: That can take on only a finite set of values Whose probabilities sum to 1 rolling a die {1,2,3,4,5,6} rolling 2 dice {2,3,4,5,6,7,8,9,10,11,12} choosing a card from a standard deck (ignoring suit) {A,2,3,4,5,6,7,8,9,10,J,Q,K}

Probability Distribution the probability distribution of a random variable X, is a function which provides the probability of each possible value of X may be represented as a table of values or graph e.g., rolling a die:

Probability Distribution for 2 Dice

What would a probability distribution graph for three dice look like? We will try it! For three dice, figure out all possible outcomes Then find out how many ways there are to create each outcome Fill in a table like the one below Outcome 3 4 5 6 7 8 9 … # ways 1

Probability Distribution for 3 Dice Outcome 3 4 5 6 7 8 9 10 # cases Outcome 11 12 13 14 15 16 17 18 # cases

Probability Distribution for 3 Dice Outcome 3 4 5 6 7 8 9 10 # cases 1 15 21 28 36 Outcome 11 12 13 14 15 16 17 18 # cases 36 28 21 10 6 3 1

So what does an experimental distribution look like? A simulated dice throw was done a million times using a computer program and generated the following data What is the most common outcome? Does this make sense?

Back to 2 Dice What is the expected value of throwing 2 dice? How could this be calculated? So the expected value of a discrete variable X is the sum of the values of X multiplied by their probabilities

Example 1a: tossing 3 coins 0 heads 1 head 2 heads 3 heads P(X) ⅛ ⅜ What is the likelihood of at least 2 heads? It must be the total probability of tossing 2 heads and tossing 3 heads P(X = 2) + P(X = 3) = ⅜ + ⅛ = ½ so the probability is 0.5

Example 1b: tossing 3 coins 0 heads 1 head 2 heads 3 heads P(X) ⅛ ⅜ What is the expected number of heads? It must be the sums of the values of x multiplied by the probabilities of x 0P(X = 0) + 1P(X = 1) + 2P(X = 2) + 3P(X = 3) = 0(⅛) + 1(⅜) + 2(⅜) + 3(⅛) = 1½ So the expected number of heads is 1.5

Combinations Recall that C(n, r) is the number of ways r objects can be chosen from n when order doesn’t matter.

Example 2a: Selecting a Committee of three people from a group of 4 men and 3 women What is the probability of having at least one woman on the team? There are C(7,3) or 35 possible teams C(4,3) = 4 have no women C(4,2) x C(3,1) = 6 x 3 = 18 have one woman C(4,1) x C(3,2) = 4 x 3 = 12 have 2 women C(3,3) = 1 has 3 women

Example 2a cont’d: selecting a committee 0 women 1 woman 2 women 3 women P(X) 4/35 18/35 12/35 1/35 What is the likelihood of at least one woman? It must be the total probability of all the cases with at least one woman P(X = 1) + P(X = 2) + P(X = 3) = 18/35 + 12/35 + 1/35 = 31/35

Example 2b: selecting a committee 0 women 1 woman 2 women 3 women P(X) 4/35 18/35 12/35 1/35 What is the expected number of women? 0P(X = 0) + 1P(X = 1) + 2P(X = 2) + 3P(X = 3) = 0(4/35) + 1(18/35) + 2(12/35) + 3(1/35) = 1.3 (approximately)

MSIP / Home Learning p. 277 #1-9, 12, 14

Warm up How many different Bloody Marys are possible? Note: Legal drinking age only. Please drink responsibly

Bloody Mary (X3) Method 1: Combinations Method 2: Counting outcomes There are 11 toppings to choose from. You can choose 0, 1, 2, 3, …, 11 from the 11. C(11,0)+C(11,1)+C(11,2)+ … + C(11, 11) = 1+11+55+165+330+462+462+330+165+55+11+1 = 2048 Method 2: Counting outcomes Each topping has 2 outcomes: yes or no 211 = 2048

5.2 Pascal’s Triangle Chapter 5 – Probability Distributions and Predictions Learning Goal: Use Pascal’s Triangle to Solve Path Counting Problems Questions? p. 277 #1-9, 12, 14 MSIP/Home Learning: p. 289 #1, 2aceg, 6-8, 11-12

How many routes are there to the top right-hand corner? you need to move North 4 spaces and East 5 spaces This is the same as rearranging the letters NNNNEEEEE It is also the same as choosing 4 of the 9 moves to be N or, choosing 5 of the 9 moves to be E.

Permutation AND Combination? As a permutation with identical objects: There are 9 moves  9! different arrangements However, 4 are identical moves N, and 5 are identical moves E 9! 4!5! =126 As a combination: There are 9 moves required _ _ _ _ _ _ _ _ _ Choose 4 to be Ns, the rest are Es  C(9, 4) = 126 Or, choose 5 to be Es, the rest are Ns  C(9, 5) = 126

Pascal’s Triangle the outer values are always 1 the inner values are determined by adding two values diagonally above 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

Pascal’s Triangle sum of each row is a power of 2 1 = 20 2 = 21 1 1 1 4 = 22 8 = 23 16 = 24 32 = 25 64 = 26 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 Uses? Binomial Expansion Combination calculator e.g., choose 2 items from 5 go to the 5th row, the 2nd number = 10 (start counting at 0) Modeling the electrons in each shell of an atom (google ‘Pascal’s Triangle electron’)

Pascal’s Triangle – Cool Stuff 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 each diagonal is summed up in the next value below and to the left called the “hockey stick” property

Pascal’s Triangle – Cool Stuff numbers divisible by 5 similar patterns exist for other numbers http://www.shodor.org/interactivate/activities/pascal1/

Pascal’s Triangle can also be seen in terms of combinations

Pascal’s Triangle - Summary symmetric down the middle outside number is always 1 second diagonal values match the row numbers sum of each row is a power of 2 sum of nth row is 2n First row is row 0 number inside a row is the sum of the two numbers above it

And one more thing… remember that for the inner numbers in the triangle, any number is the sum of the two numbers above it for example 4 + 6 = 10 this suggests the following: which is an example of Pascal’s Identity

For Example…

How can this help us solve our original problem? 1 5 15 35 70 126 4 10 20 56 3 6 21 2 so by overlaying Pascal’s Triangle over the grid we can see that there are 126 ways to move from one corner to another

How many routes pass through the green square? to get to the green square, there are C(4,2) ways (6 ways) to get to the end from the green square there are C(5,3) ways (10 ways) in total there are 60 ways

How many routes do not pass through the green square? there are 60 ways that pass through the green square there are C(9,5) or 126 ways in total then there must be 126 – 60 = 66 paths that do not pass through the green square

Path Counting – off the grid How many paths are there through the word BREAK? B B B B R R R E E E E A A A K K

Warm up How many ways are there to spell SUCCESS? (moving only along diagonals) S U U C C C C C E E E S S

Warm up How many ways are there to spell SUCCESS? S18 S9 S9 E3 E6 E3 C1 C2 C1 U1 U1 S1

MSIP / Home Learning Read the examples on pp. 281-287 The example starting on the bottom of p. 287 is important p. 289 #1, 2aceg, 6-8, 11-12