1. Minds on! CP Canoe Club Duck Derby Prizes
Maximum of 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?

2. Expected Value Calculation
For tickets
E(X) =(1/2000)(2000) + (1/2000)(600) + (1/2000)(250)
= 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

3. 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?

4. Basket Draw – what I won $150 Miken softball glove (sold)
1 night at Qualify Suites London (gifted)
$55 at Fox and Firkin, Toronto (gifted)
$ % 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. 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. 7-day weather forecast What is POP? Probability of Precipitation
What does it mean?

8. 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

9. 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}

10. 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:

11. Probability Distribution for 2 Dice

12. 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

13. 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

14. 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

15. 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?

16. 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

17. 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

18. 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

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

20. 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

21. 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 + 1/35 = 31/35

22. 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)

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

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

25. 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) =
= 2048
Method 2: Counting outcomes
Each topping has 2 outcomes: yes or no
211 = 2048

26. 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

27. 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.

28. 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

29. Pascal’s Triangle
the outer values are always 1
the inner values are determined by adding two values diagonally above 1
1 1

30. 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

31. Pascal’s Triangle 1
1 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’)

32. Pascal’s Triangle – Cool Stuff1
1 1
each diagonal is summed up in the next value below and to the left
called the “hockey stick” property

33. Pascal’s Triangle – Cool Stuff
numbers divisible by 5
similar patterns exist for other numbers

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

35. 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

36. 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 = 10
this suggests the following:
which is an example of Pascal’s Identity

37. For Example…

38. 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

39. 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

40. 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

41. 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

42. 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

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

44. 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