AP Statistics6.1-6.2 Probability Models

Understand the term “random”

Implement different probability models

Use the rules of probability in calculations

Learning Objective:

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.

Random

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.

Probability

  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

6.2 Probability Models

x H TP(x) ½ ½

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

Probability Models

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….

Example:

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

Tree Diagram-

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

Multiplication Principle-

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 =

160 1 2 3 4

TTTT HTTT HHTT THHH HHHH

THTT HTHT HTHH

TTHT HTTH HHTH

TTTH THHT HHHT

TTHH

THTH

X 0 1 2 3 4

P(x) 1/16 1/4 3/8 1/4 1/16

What is the probability Distribution?

Ex: Generate a random decimal number. What is the sample space?

S={all numbers between 0 and 1}

Example

a) S= {G,F}

b) S={length of time after treatment}

c) S={A,B,C,D,F}

Pg. 322: 6.9

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

Probability Rules

#3-

#4- Disjoint- A and B have no outcomes in common (mutually exclusive)

P(A or B)= P(A) + P(B)

Probability Rules

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 ∅

Venn Diagram:

P(A)= 0.34 P(B)=0.25 P(A ∩ B)=0.12

Display the probabilities by using a Venn Diagram.

What is the sum of these probabilities?1

P(not married)=1- P(M)= 1 – 0.574 = 0.426

P(never married or divorced)= 0.353 + 0.071 = 0.424

Marital StatusMarital Status

NeverMarried

Married Widowed Divorced

Probability 0.353 0.574 0.002 0.071

 

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

Benford’s LawFirst Digit

1 2 3 4 5 6 7 8 9

Prob. .301 .176 .125 .097 .079 .067 .058 .051 .046

D={first digit is not 1}P(D)= 1- 0.301= 0.699

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

Benford’s Law Cont.

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

Equally likely outcomes

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

Pg. 330: 6.18, 6.19

Rule 5:

P(A and B)= P(A) P(B)(only for independent events!)

The Multiplication Rule for Independent Events

6.24: One Big: 0.63 small: (0.8)³=0.512

6.25: (1-0.05)^12=0.5404

6.26: the events aren’t independent

Pg. 335