1. 6.3/6.4 Probability

2. Would you take the bet? Why or why not?
Trent and Malcolm are spectators at a craps table in Las Vegas. Trent offers the following bet to Malcolm:
When the dice (two die) are tossed, Trent will pay Malcolm $5 if the two-dice sum is 6, 7, 8, or 9, while Trent will win $6 from Malcolm if the sum is 2, 3, 4, 5, 10, 11, or 12. Trent reasons that “He has only four winning sums, while I have 7; and I win $1 more when I win. This bet is sure thing!”

3. The beginning of Probability
A 17th century French nobleman, gambler, and general rogue enjoyed dice games.
Game 1 involved rolling a die four times. If a six ever appeared, he won the bet; if not, he lost. He lost frequently, but seemed to come out ahead in the long run.
Game 2 involved rolling 2 dice 24 times. He would win if the sum of 12 ever happened.
He thought Game 2 was equivalent to Game 1 because the chance of winning is 1/6. The probability of winning on Game 2 is 1/36. So, if you roll the dice six more times than the first game, then it will fix the disparity.
Was he correct in his thinking??
Perplexed that he was wrong, he asked Pierre de Fermat to work on the problem, and this is how modern probability was started.

4. Random experiments Examples of random experiments:
The color of a child’s hair or eyes before birth
Knowing your SAT score before you take it
Treating cancer with radiation—how many cancer cells will it kill?
Knowing the yield of a corn field before you plant it.
A random experiment is a repeatable process whose out comes depend on chance; the word “random” serves only to indicate the unpredictability of the outcome.
Other examples: Tossing a coin, Rolling a die, Choosing a card, etc.

5. Sample Space
A sample space, denoted by S, is a set whose elements describe all possible outcomes in an experiment.
Example: What is the sample space of tossing coin?
𝑆={𝐻, 𝑇}
Example: What is the sample space of rolling a die?
𝑆={1, 2, 3, 4 ,5, 6}
Example: What is the sample space of choosing a card from the deck?
𝑆={𝑐𝑙𝑢𝑏 2, 𝑐𝑙𝑢𝑏 3, …, 𝑐𝑙𝑢𝑏 𝐴, ℎ𝑒𝑎𝑟𝑡 2,…𝑠𝑝𝑎𝑑𝑒 2, …,𝑑𝑖𝑎𝑚𝑜𝑛𝑑 2,…}

6. Finite versus Infinite Sample Spaces
The last three examples were finite sample spaces. We can count the number of possibilities.
Suppose we are recording the amount of time it takes for a light bulb to burn out. This is an infinite sample space because there are infinite amount of numbers between 0 and however long it takes.

7. Example
Suppose you are picking 2 spark plugs at random from the 6 that are in your car. What is the sample space? Number the spark plugs How many should be in your sample space? List it out…
1,2 1,3 1,4 1,5 1,6 2,1 2,3 2,4 2,5 2,6 3,1 3,2 3,4 3,5 3,6
4,1 4,2 4,3 4,5 4,6 5,1 5,2 5,3 5,4 5,6 6,1 6,2 6,3 6,4 6,5

8. Events
Any collection of outcomes (subset of the sample space) is called an event.
An event with only one outcome is called a simple event.
An event containing two or more outcomes is called a compound event.
Events are represented by italic capital letters A, B, C, etc.

9. Example: think back to the spark plugs
Let’s modify the problem by supposing that out interest lies only in whether the spark plug is good or defective. What is the sample space? Drawing a tree diagram could help by find this…
𝑆={𝐺𝐺, 𝐺𝐷, 𝐷𝐺, 𝐷𝐷} G
Consider the following Events of this problem:
A: Exactly one defective plug
B: At least 1 defective plug
C: At most one defect. plug
F: The plugs test the same
H: The second plug is defect.
J: One plug is good G D
A={DG, GD}
B={DG, GD, DD}
C={DG, GD, GG}
F={DD, GG}
H ={DD, GD}
J={DG, GD} Go G D D

10. Example Consider the former sample space: 𝑆={𝐺𝐷, 𝐺𝐺, 𝐷𝐺, 𝐷𝐷}
List the subset that defines the event:
The number of defective plugs is one more than the number of good plugs ∅
b. The second plug is defective
{𝐺𝐷, 𝐷𝐷}
c. The first plug is good
{𝐺𝐺, 𝐺𝐷}

11. Example
Matt and Donna are an hour’s drive from home and cannot remember if they locked all three of their doors (front, back, and side). They call the neighbor to have her check all of the doors. What is the sample space?
𝑆={𝑂𝑂𝑂, 𝑂𝑂𝐿, 𝑂𝐿𝑂, 𝑂𝐿𝐿, 𝐿𝑂𝑂, 𝐿𝑂𝐿, 𝐿𝐿𝑂, 𝐿𝐿𝐿}

12. Example continued Consider the last sample space:
𝑆={𝑂𝑂𝑂, 𝑂𝑂𝐿, 𝑂𝐿𝑂, 𝑂𝐿𝐿, 𝐿𝑂𝑂, 𝐿𝑂𝐿, 𝐿𝐿𝑂, 𝐿𝐿𝐿}
Write out each of the following events:
A: One door is locked
B: At least one is locked
C: Two are locked
D: At least two are locked
E: The back door is locked
F: The side door is open.
G: At most two are locked.
H: At most one is open
I: All three are open
𝐴= 𝐿𝑂𝑂, 𝑂𝐿𝑂, 𝑂𝑂𝐿
𝐵={𝑂𝑂𝐿, 𝑂𝐿𝑂, 𝑂𝐿𝐿, 𝐿𝑂𝑂, 𝐿𝑂𝐿, 𝐿𝐿𝑂, 𝐿𝐿𝐿}
𝐶={𝑂𝐿𝐿, 𝐿𝑂𝐿, 𝐿𝐿𝑂}
𝐷={𝐿𝐿𝑂, 𝑂𝐿𝐿, 𝐿𝑂𝐿, 𝐿𝐿𝐿}
𝐸={𝑂𝐿𝑂, 𝐿𝐿𝑂, 𝐿𝐿𝐿}
𝐹={𝑂𝑂𝑂, 𝑂𝑂𝐿, 𝐿𝑂𝐿, 𝐿𝑂𝑂}
𝐺={𝑂𝑂𝑂, 𝑂𝑂𝐿, 𝑂𝐿𝑂, 𝑂𝐿𝐿, 𝐿𝑂𝑂, 𝐿𝑂𝐿, 𝐿𝐿𝑂}
𝐻={𝑂𝐿𝐿, 𝐿𝑂𝐿, 𝐿𝐿𝑂, 𝐿𝐿𝐿}
𝐼={𝑂𝑂𝑂}

13. Example Continued
Use the events from the previous slide to answer the following: 𝑆={𝑂𝑂𝑂, 𝑂𝑂𝐿, 𝑂𝐿𝑂, 𝑂𝐿𝐿, 𝐿𝑂𝑂, 𝐿𝑂𝐿, 𝐿𝐿𝑂, 𝐿𝐿𝐿}
𝐴∪𝐵
𝐴∩𝐵
𝐴∪𝐵 𝑐
𝐴 𝑐 ∩ 𝐵 𝑐
𝐴 𝐶
𝐴= 𝐿𝑂𝑂, 𝑂𝐿𝑂, 𝑂𝑂𝐿
𝐵={𝑂𝑂𝐿, 𝑂𝐿𝑂, 𝑂𝐿𝐿, 𝐿𝑂𝑂, 𝐿𝑂𝐿, 𝐿𝐿𝑂, 𝐿𝐿𝐿}
𝐶={𝑂𝐿𝐿, 𝐿𝑂𝐿, 𝐿𝐿𝑂}
𝐴∪𝐵=𝐵= 𝑂𝑂𝐿, 𝑂𝐿𝑂, 𝑂𝐿𝐿, 𝐿𝑂𝑂, 𝐿𝑂𝐿, 𝐿𝐿𝑂, 𝐿𝐿𝐿
𝐴∩𝐵= 𝐿𝑂𝑂, 𝑂𝐿𝑂, 𝑂𝑂𝐿
𝐴∪𝐵 𝐶 ={𝑂𝑂𝑂}
𝐴 𝐶 ∩ 𝐵 𝐶 ={𝑂𝑂𝑂}
𝐴 𝐶 ={𝑂𝑂𝑂, 𝑂𝐿𝐿, 𝐿𝑂𝐿, 𝐿𝐿𝑂, 𝐿𝐿𝐿}

14. homework
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