1. Minds on!
If you choose an answer to this question at random, what is the probability you will be correct? A) 25% B) 50% C) 100% D) 25%

2. Steps to Solving Probability Problems
G = Given: List knowns, draw diagram / table / list
R = Required to find: Identify the unknown, look for keywords
A = Analyze: Write the formula
S = Solve: Plug knowns into formula and compute
P = Present: State your answer using a sentence

3. Warm up #1
In , the Toronto Maple Leafs won 19 of their first 50 games. Their leading goal scorer, Leo Komarov, scored at least one goal in 10 of those. Given that the Leafs won their next game, what is the probability that Komarov scored a goal?

4. Solution Let W be the event that the Leafs won.
Let K be the event that Leo Komarov scored.
The probability that Komarov scored, given that the Leafs won is 0.53.

5. Warm up #2 Let H1 be the event that the first card is a heart
Find the probability that two hearts are dealt consecutively from a standard deck (no replacement).
Let H1 be the event that the first card is a heart
Let H2 be the event that the second card is a heart
P(H1 ∩ H2) = P(H2 | H1) x P(H1)
= 12/51 x 13/52
= 156/2652
= 0.06

6. 4.5 Finding Probability Using Tree Diagrams and Outcome Tables
Learning goal: Calculate probabilities using tree diagrams and outcome tables
Questions? pp. 235 – 238 #1, 2, 4, 6, 7, 9, 10, 19
MSIP / Home Learning: pp. 245 – 249 #2, 3, 5, 7, 9, 12, 13a, 14

7. Tree Diagram 1
if you flip a coin twice, you can model the possible outcomes using a tree diagram or an outcome table resulting in 4 possible outcomes
Flip 1
Flip 2
Simple
Event H HH T HT TH TT H T T
Toss Toss 2

8. How to Draw a Tree Diagram
Each branch represents an event
Draw the branch for the first event
Add a branch to each for the second event
Repeat for additional events
Follow each branch to list the outcomes

9. Tree Diagram 2
if you roll 1 die and then flip a coin there are 12 possible outcomes H T
(1,H) 1
(1,T) H T
(2,H) 2
(2,T) H T
(3,H) 3
(3,T)
(4,H) H T 4
(4,T) H T
(5,H) 5
(5,T)
(6,H) H T 6
(6,T)

10. Tree Diagram 2 (cont’d)
The sample space is all ordered pairs in the form (d,c), where
d represents the roll of a die
c represents the flip of a coin
e.g., (4,T) represents rolling a 4 then flipping a tail
There are 6 x 2 = 12 possible outcomes
What is P(odd roll, head) = ?
There are 3 possible outcomes for an odd die and a head: (1, H), (3, H) and (5, H)
So the probability is 3/12 or ¼

11. Multiplicative Principle for Counting
The total number of outcomes is the product of the number of possible outcomes at each step in the sequence
If a is selected from A, and b selected from B…
n(a,b) = n(A) x n(B)
This assumes that each outcome has no influence on the next outcome

12. Examples Pt 1 How many outcomes are there when rolling 2 dice?
How many sets of initials are there?
26 x 26 = 676
What if letters cannot be repeated?
This adds a restriction on the 2nd letter
26 x 25 = 650

13. Examples Pt 2 Airport codes are arrangements of 3 letters
e.g., YOW=Ottawa, YYZ=Toronto
How many are possible?
26 choices for each of the 3 positions
26 x 26 x 26 =
How many with no repeated letters?
26 x 25 x 24 =

14. Examples Pt 3
How many possible postal codes are there in Canada? The format is LDL DLD
26 x 10 x 26 x 10 x 26 x 10 =
263 x 103
How many are in Ontario where the 1st letter is K, L, M, N or P
5 x 10 x 26 x 10 x 26 x 10 =
What if there are no repeated characters?
5 x 10 x 25 x 9 x 24 x 8

15. Examples Pt 4 How many 4-digit PINs are possible on a smartphone?
10x10x10x10 =
What if you saw the first character pressed?
10x10x10 = 1000
If you saw the first character, what is the probability of guessing the PIN correctly?
1 / 1000

16. Minds on! Suppose Mr. Lieff installs a 4-digit capcha on his website
How many different answers are possible?
10x10x10x10 =
What if no numbers can be repeated?
10x9x8x7 = 5 040
What if you know the first digit is 3?
1x10x10x10 = 1 000
What is the probability of guessing each of these?

17. Independent and Dependent Events
Independent events: the occurrence of one event does not change the probability another
Coins, dice, cards w/replacement, spinners are all independent
Ex. What is the probability of getting heads when you have thrown an even die?
These are independent events, because knowing the outcome of the first does not change the probability of the second

18. Multiplicative Principle for Probability of Independent Events
If we know that A and B are independent events, then…
P(B | A) = P(B)
Knowing A occurred does not affect P(B)
We can also prove that if two events are independent the probability of both occurring is…
P(A and B) = P(A) × P(B)

19. Example 1 A sock drawer has a red, a green and a blue sock
You pull out one sock, replace it and pull another out
a) Draw a tree diagram representing the possible outcomes
b) What is the probability of drawing the red sock both times?
These are independent events R B G

20. Example 2
If you draw a card, replace it and draw another, what is the probability of getting two aces?
4/52 x 4/52 = 16/2704 = 1/169 = 0.006
These are independent events
What if the first card is not replaced?
4/52 x 3/51 = 12/2652

21. Example 3 - Predicting Outcomes
What if the outcomes are not equally likely?
Mr. Lieff is playing Texas Hold’Em
He finds that he wins 70% of the pots when he does not bluff
He also finds that he wins 50% of the pots when he does bluff
If there is a 60% chance that Mr. Lieff will bluff on his next hand, what are his chances of winning the pot?
We will start by creating a tree diagram

22. Tree Diagram 0.5 Win pot P=0.6 x 0.5 = 0.3 bluff 0.6 0.5 Lose pot
0.7
P=0.4 x 0.7 = 0.28
0.4
no bluff
0.3
Lose pot
P=0.4 x 0.3 = 0.12

23. Continued… P(no bluff, win) = P(no bluff) x P(win | no bluff)
P(bluff, win) = P(bluff) x P(win | bluff)
= 0.6 x 0.5 = 0.30
Probability of a win: = 0.58
So Mr. Lieff has a 58% chance of winning the next pot

24. MSIP / Home Learning Read the examples on pages 239-244
Complete pp. 245 – 249 #2, 3, 5, 7, 9, 12, 13a, 14