1. Unit 6: Probability

2. Uses of Probability Probability is used all of the time in real life
Gambling
Sports
Weather
Insurance
Medical Decisions
Standardized Tests
And others

3. Definition of Probability
“The likelihood of something happening”
We use this to answer questions like:
What is the chance of rain tomorrow?
Will you win at Black Jack?
Who will win the Super Bowl?
Is the answer A, B, C, or D?
What are the chances of rolling a 13 with 2 dice?

4. Basic Probability
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐸𝑣𝑒𝑛𝑡= # 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑓𝑢𝑙 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 # 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 Probability is always between 0 and 1 .5

5. Random Things to Know
Dice (singular = “die”) Most cases: 6 sided Numbers 1,2,3,4,5,6 𝑃 𝐴 = 1 6 Special Cases: 4 sided 8 sided 10 sided 12 sided 20 sided

6. Random Things to Know
Cards Typical Deck: 52 cards 4 Suits (13 cards each) Clubs Spades Face 3 Face 1 Ace 1 Ace Hearts Diamonds 1 Ace 1 Ace

7. Random Things to Know License Plates 7 Characters
Different states have different rules
𝑃 𝑙𝑒𝑡𝑡𝑒𝑟𝑠 = 1 26
𝑃 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 = 1 10

8. Counting Principle
If you have 6 shirts and 3 pants how many different outfits can you create?

9. Counting Principle
If you have M of one option and you have N of another option, Then there are 𝑴∙𝑵 ways of doing both M = shirts N = pants Number of outfits you can make = 𝑀∙𝑁 6∙3=18

10. Counting Principle
When flipping a coin 15 times how many results are possible? *Think how many different results are there when you flip a coin*

11. Counting Principle A restaurant has on its menu
5 choices for appetizers
3 choices for main course
2 choices for dessert
How many different meals (appetizer, main course, and dessert) can you choose?

12. Factorials!

13. Multiplication Rule and Addition Rule of Probability

14. Multiplication Rule and Addition Rule of Probability
Examples What is the probability of rolling a 6 on a die and then rolling another 6? 𝑃 6 = 1 6 𝑃 6 = 1 6 so 𝑃 𝐴 𝑎𝑛𝑑 𝐵 = 1 6 ∙ 1 6 = 1 36 What is the probability rolling a 1 or a 2 on a single roll of a die? 𝑃 1 = 1 6 𝑃 2 = 1 6 so 𝑃 𝐴 𝑜𝑟 𝐵 = = 1 3

15. BIG QUESTION: DOES ORDER MATTER??
Combinations
If you have 5 trophies but only space on a shelf for 2 of them how many different ways can you arrange your trophies?
BIG QUESTION: DOES ORDER MATTER??

16. Combinations
Remember the Counting Principle: M*N = total number of ways to select items How many trophies can you choose between? 5 How many spots are there? 2 So… 5∙2=10

17. Combinations
Def: A way of selecting items where the order does not matter If you order pizza it doesn’t matter if you tell them “Peperoni, Pineapple, and Sausage” or “Sausage, Peperoni, and Pineapple” NO! It all goes on the pizza! The order doesn’t matter

18. Combinations 𝑛 ∁ 𝑟 𝑜𝑟 𝑛 𝑟 = 𝑛! 𝑟! 𝑛−𝑟 ! n = total number of elements
𝑛 ∁ 𝑟 𝑜𝑟 𝑛 𝑟 = 𝑛! 𝑟! 𝑛−𝑟 !
n = total number of elements
r = number of items chosen
𝑛 ∁ 𝑟 = 5! 2! 5−2 ! = 120 2(6) =10

19. Permutations
You and your 3 friends are standing in line to buy tickets to a movie. How many ways are there for you to arrange yourselves?

20. Permutations Remember the Counting Principle:
M*N = total number of ways to select items
How many choices do you have for the first spot? (4)
How many choices do you have for the second spot? (3)
How many choices do you have for the third spot? (2)
How many choices do you have for the fourth spot? (1)
So 4∙3∙2∙1=24

21. Permutations
Def: A way of selecting items where the order does matter In races who comes in 1st, 2nd, and 3rd is very important for prizes, and rankings. The order does matter.

22. Permutations If repetition is not allowed: 𝑛 𝑃 𝑟 = 𝑛! 𝑛−𝑟 !
If repetition is allowed: 𝑛 𝑃 𝑟 = 𝑛 𝑟
n = total number of elements
r = number chosen
𝑛 𝑃 𝑟 = 𝑛! 𝑛−𝑟 ! = 4! 4−4 ! = 24 1 =24