1. What is probability? Simply put, it’s the
chance of an event occurring
Definitions:
Probability experiment  A process that leads to welldefineed results, or outcomes. Outcome  The result of a single trial of a probability experiment Sample space  The set of all possible outcoomes of a probability experiment Event  One or more outcomes of a probability experiment

Probability  3 types
Classical, Empirical, and Subjective

Type 1: Classical
 Uses sample spaces to determine the numerrical probability that an event
will occur
a. Assumes that all outcomes in the sample
space are equally probable to occur
Examples:
Formula for Classical Probability: P(E) = n(E)/n(S) where:
**Notes: 1. Probabilities are generally
expressed as percentages, decimals, or fractions
I realize that the terms "n(E) = number of outcomes in event E" and " n(S) = total # of outcomes in sample space S” sound quite strange. Look at it this way: Pretend there is a box (see below) with 40 colored
marbles (13 red, 12 blue, 9 orange, 6 green).
1. The n(S) ( the total # of outcomes in sample
space “S”) is 40.
Now, I want to know the probability of getting either a red ball or a blue ball when I pick one at random. 1. The n(S) ( the total # of outcomes in sample
space “S”) is 40.

Complementary Events
The complement of an event "E" is the set of outcomes in the sample space that are not included in the outcomes of the event E, and are denoted as E. Rule for Complementary Events P(E) = 1  P(E) or P(E) = 1 P(E) ** aka : If the probability of an event or its
complement is known, then the other can be found by subtracting the known
probability from 1

Type 2: Empirical
Probability  Relies on actual
experience or experiments to determine the likelihood (probability) of
outcomes
What if someone weighted the dice (CHEATER!), and you found out about it? You watch the dice for a while, and find out that "2" is the preferred number? You watch the die for a while (say 100 throws) and "2" come up over half the time (you keep count). This is found out by experience  EMPIRICISM  or Empirical probabilty! Formula for Empirical probability: P(E) = f/n where:
**Note: There's another side of this, though. While the probability of getting a ‘heads’ when flipping a coin once is exactly 1/2, when it’s tossed 100 times it may not come up heads exactly 50 times (due largely to chance variation). It's possible that through simple good fortune, heads may come up 55 times out of 100  this might trick someone into thinking that there is a preference for heads. However, as the number of trials increases, the empirical probability of getting a ‘heads’ will approach 1/2, if the coin is balanced correctly. This phenomenon is known as The Law of Large Numbers
and will hold true for any type of gambling game.

Type 3  Subjective Probability
 a probability value based on an educated guess or estimate, employing
opinion and inexact information
Examples: Doctor says there’s a 30% chance a man has cancer
No equations! This is an educated guess, based on expertise, qualitative experience (not empirical), etc. 
The Addition Rules for
Probability
Many problems involve finding the probability of two or more events occurring. There are different ways of determining probability, based on whether the events are mutually exclusive. ** Two events are considered mutually exclusive if they cannot occur at the same time (i.e., they have no outcomes in common)
Addition Rule 1 When two events A and B are mutually exclusive, the probability that either A or B will occur is given by: P(A or B) = P(A) + P(B)
** But what if the events are NOT mutually exclusive?
Addition Rule 2  If events A and B are NOT mutually exclusive, then the probability that either A or B will occur is given by:
P(A or B) = P(A) + P(B)  P(A and B)
** Note  the addition rules can be extended for 3 or more events in the following ways:
Addition Rule 1 (mutually exclusive) P(A or B or C) = P(A) + P(B) + P(C)
Addition Rule 2 (not mutually exclusive)
P(A or B or C) =
Addition Rule 2 (not mutually exclusive) OUCH!!!!!! WOW! What are you saying? WHAT IS THIS?!?!?!?!? OK, this is going to be a long explanation  get a
cup of coffee.....
START OF Explanation for Addition Rule 2 (not mutually exclusive) Look at this graphic...
This figure shows the Black Box "D", which contains the numbers  1 through 25. Since Box "D" contains all the numbers  1 through 25  it also includes the numbers in Circle "A", Circle "B" and Circle "C". Notice:
Here the question: What is the probabilty of
getting a number that is in "A", "B", or "C" (assuming a random pick)?
But WAIT! The numbers "17", "18", "19", "20", "21", "22", "23", "24" , and "25" are NOT IN "A", "B", or "C"! That's 9 out of 25 total numbers  9/25, or 0.36 (or 36%). Now get this: These numbers represent the COMPLEMENT of our intended goal (picking a number that IS IN "A", "B", or "C"! So, the actual probabilty of getting a number that is in "A", "B", or "C" (assuming a random pick) is : 10.36 = 0.64
So, we know the REAL ANSWER is 64%. What happened? Well, look at the graphic again.
We counted the probabilty of getting a number from Circle "A".
Then, we counted the probabilty of getting
a number from Circle "B".
PROBLEM: when we counted "A", we already included "1" and "3". When we counted "B", we included "1" and "3" again  in ERROR! Then, we counted the probabilty of getting a number from Circle "C". PROBLEM: when we counted "B", we already included "3" and "8", and "9". When we counted "C", we included "3", "8" and "9" again  in ERROR! HOW DO WE FIX THIS? Simple: SUBTRACT ALL THE DOUBLE COUNTED ITEMS AT LEAST
ONCE!
For the point when we counted "A" and "B", we counted "1" and "3" twice. So, let's subtract out the probability of getting a number that is in BOTH "A" and "B". This is:  P(A and B) This is from the equation for Addition
Rule 2 (not mutually exclusive)  above.
For the point when we counted "A" and "C", we counted "2", "3", and "4" twice. So, let's subtract out the probability of getting a number that is in BOTH "A" and "C". This is:
 P(A and C)
For the point when we counted "B" and "C", we counted "3", "8", and "9" twice. So, let's subtract out the probability of getting a number that is in BOTH "A" and "C". This is:
 P(B and C)
BUT WAIT!!! We added in the number "3" a total of three times, then subtracted it three times! so we did not actually include "3" yet! The problem here is that "3" is in circles "A", "B", AND "C"! So, then, let's ADD "3" back in by adding in the probability of getting a number that is in "A" and "B" AND "C". This is: +
P(A and B and C)
So, here's the calculation: P(A or B or C) =
END OF Explanation
for Addition Rule 2 (not mutually exclusive)

I hope that made sense!
:)
Multiplication Rules and
Conditional Probability
Used to find the probability of two or more events that occur in sequence. Independent events  Two events A and B are said to
be independent if the fact that A occurs does not affect the probability
of B occurring.
Example: Two die are rolled separately (and one does not hit the other). That fact that the first was a "1" does not affect the outcome of the other die  it could be 1,2,3,4,5, or 6.
Multiplication Rule 1  When two events are independent, the probability of BOTH of them occurring is given by: P(A and B) = P(A) P(B) or , this is the also stated as: P(A and B) = P(A) x P(B) In English: The probability of both "A" and
"B" happening at the same time is equal to the probability of "A" occurring,
TIMES the probability of "B" occuring.
We can also extend this rule to 3 or more events: P(A and B and C and ... and K) = P(A) P(B) P(C) ... P(K)
But what if the events under consideration aren’t independent? Dependent events  When the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability is changed, then the events are said to be dependent. **This brings up the concept of Conditional Probability
The probability of an event B occurring given that an event A has already occurred is called conditional probability and has the notation: P(BA): “the probability that B occurs
given that A already occurred”
Huh? OK, let's play a card game. Pick a card  any card. First one to get an ACE Wins! 1. There are 52 cards in the deck (NO JOKERS!)
So, the previous picking of a card AFFECTS your chances of SUCCESS! This is "Conditional Probability".
Multiplication Rule 2  When two events are dependent, the probability of both occurring is given by: P(A and B) = P(A) P(BA) For 3 or more events: P(A and B and C) = P(A) P(BA)
P(CB)
Remember we found that multiplication rule 2 was given by: P(A and B) = P(A) P(BA) We can use this formula and simple math to find the conditional probability! Divide through both sides by P(A) to get: P(A and B)/P(A) = P(BA) or rewritten, this becomes: P(BA) = P(A and B)/P(A) the “Formula for Conditional Probability” I realize that "P(BA)" looks odd. We say this as "The probabilty of 'B' taking place , given that 'A' just took place". 
Probabilities involving
the phrase “at least”
When dealing with problems of this nature (see examples 535, 36 & 37), it is usually easier to find the probability when the event in question doesn’t happen at all. This is then the complementary to “at least”, and we know from before that: Rule for Complementary Events P(E) = 1  P(E) or P(E) = 1 P(E) ** aka : If the probability of an event or its complement is known, then the other can be found by subtracting the known probability from 1 So simply determine the probability for the event not
occurring, then subtract that probability from 1 to get the probability
of the “at least” case.

Probability and Counting
Techniques
We often need to use a combination of techniques to determine probabilities in certain situations. Using the counting rules described in Chapter 4, plus the probability rules in this chapter, we can find probabilities for more complex problems.

HOMEWORK:
No reading assignment. Go to Lesson 6. 