Probability Models 6.2

First—A little review  In your notes, from memory— (Self- check quiz) Define Sample Space Write down at least one rendering of the Law of Large Numbers What’s the difference between Classic and Empirical Probability?

The Fundamental Counting Principle (AKA, the Multiplication Principle)  “The number of possible final outcomes is the product of the number of options at each decision point.”  Tree Diagrams, in other words  (Example on board)

Replacement vs. Nonreplacement  Classic Probability problem: Bag of marbles A bag contains 10 marbles. 6 are blue, 4 are red. Find the probability of pulling out two red marbles with replacement Find the probability of pulling out two red marbles without replacement

The Rules of Probability(See page. 343) 1.All probabilities are between 0 and 1, inclusive No negative probabilities! (“…something’s rotten in Denmark…”) Probabilities can be expressed as a decimal (best), a fraction (just as good), or a percent (CAUTION)

The Rules of Probability(See page. 343) 2.The sum of all probabilities for a given sample space is 1

The Rules of Probability(See page. 343) 3.The Complement Rule: The probability of not A is 1 – the probability of A. P(~A) = 1 - P(A) Symbology differs! If the probability of winning is.4, the probability of losing is.6

The Rules of Probability(See page. 343) 4.The Addition rule for Disjoint Events: (Also known as the OR Rule) Def: Disjoint -- “Nothing in common” -- “Mutually Exclusive” -- “Cannot occur simultaneously”

Disjoint Events  Dartboard: You cannot hit a 3 and a 20 at the same time, so they are disjoint.  Football: You cannot simultaneously win and lose. Disjoint. (Fun with Words! A Pyrrhic Victory is a situation in which you both win and lose.)Pyrrhic Victory  Dice plus Cards: You roll a three and draw an ace. Not disjoint! You can indeed do both simultaneously. One does not preclude the other.

The Rules of Probability(See page. 343) 4.The Addition rule for Disjoint Events: (Also known as the OR Rule) If two events are disjoint, then the probability of A or B = Prob (A) + Prob (B) Just add the probabilities together.

Example 1 In a game of darts, throwing a bullseye has a probability of.05 (for you!) Throwing a 20 has a probability of.10 (Also—just for you!) What’s the probabilty of hitting either the bullseye or the 20? P(bull or 20) = P(bull) + P(20) = =.15

Set Notation: OR means Union of P(Bull or 20) = P(Bull  20) = =.15 Another way: The union of the events ‘hit the bullseye’ and ‘hit the 20’ has a probability of Bull.05 + =

Venn diagrams  The outer box represents Sample Space.  Total Area = 1.0  Disjoint events are shown as non-overlapping blobs. They have no area in common! Bull.05

Venn diagrams  What’s the probability of not hitting a Bull or a 20? (IOW, how much area is outside of the yellow blobs?)  What rule does this illustrate? Bull.05