Probability Models

 Understand the term “random”  Implement different probability models  Use the rules of probability in calculations

 Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run  What does that mean to you? the more repetition, the closer it gets to the true proportion

 - if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. ◦ 1- you must have a long series of independent trials ◦ 2- probabilities imitate random behavior ◦ 3- we use a RDT or calculator to simulate behavior.

 The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. That is, the probability is long- term relative frequency.

 What is a mathematical description or model for randomness of tossing a coin?  This description has two parts.  1- A list of all possible outcomes  2- A probability for each outcome xHT P(x)½½

 Sample space S- a list of all possible outcomes.  Ex: S= {H,T} S={0,1,2,3,4,5,6,7,8,9}  Event- an outcome or set of outcomes (a subset of the sample space)  Ex: roll a 2 when tossing a number cube

 If we have two dice, how many combinations can you have? 6 * 6 = 36  If you roll a five, what could the dice read? (1,4) (4,1) (2,3) (3,2)  How can we show possible outcomes? list, tree diagram, table, etc….

 Resembles the branches of a tree. *allows us to not overlook things

 If you can do one task in a number of ways and a second task in b number of ways, then both tasks can be done in a x b number of ways.  Ex: How many outcomes are in a sample space if you toss a coin and roll a dice? 2 * 6 = 12

 Ex: You flip four coins, what is your sample space of getting a head and what are the possible outcomes? S= {0,1,2,3,4} Possible outcomes: 2 * 2 * 2 * 2 = TTTTHTTTHHTTTHHHHHHH THTTHTHTHTHH TTHTHTTHHHTH TTTHTHHTHHHT TTHH THTH

X01234 P(x)1/161/43/81/41/16

 Ex: Generate a random decimal number. What is the sample space?  S={all numbers between 0 and 1}

a) S= {G,F} b) S={length of time after treatment} c) S={A,B,C,D,F}

 With replacement- same probability and the events remain independent  Ex:  Without replacement- changes the probability of an event occurring  Ex:

 #1) 0 ≤ P(A) ≤ 1  #2) P(S) = 1

 #3-  #4- Disjoint- A and B have no outcomes in common (mutually exclusive) P(A or B)= P(A) + P(B)

 Union: “or” P(A or B) = P(A U B)  Intersect: “and” P(A and B) = P(A ∩ B)  Empty event: (no possible outcomes) S={ } or ∅

 P(A)= 0.34  P(B)=0.25  P(A ∩ B)=0.12

 What is the sum of these probabilities? 1  P(not married)= 1- P(M)= 1 – =  P(never married or divorced)= = Marital Status Never Married WidowedDivorced Probability

 A= {first digit is 1} P(A)=.30  B= {first digit is 6 or greater} P(B)=.222  C={first digit is greater than 6} P(C)=.155 First Digit Prob

 D={first digit is not 1} P(D)= =  E={1st number is 1, or 6 or greater} P(E)=0.522  F={ODD} P(F)=0.609 G={odd or 6 or greater} P(G)=0.727

 If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has probability 1/k. The probability of any event A is:  P(A)= count of outcomes in A count of outcomes in S

 Try 6.18 with your partners  A) 0.04  B) 0.69  Try 6.19  A) 0.1  B) 0.3  C) regular: 0.5 peanut: 0.4

 Rule 5: P(A and B)= P(A) P(B) (only for independent events!)

6.24: One Big: small: (0.8)³= :(1-0.05)^12= :the events aren’t independent