Chapter 4: Introduction to Probability

Permutation examples: picking a team captain, pitcher and shortstop for a baseball game -picking first, second and third place winners Combinations examples: picking three team members from a group -picking three winners

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3 ways to organize and visualize joint events

1. contingency table 2. tree diagram 3. venn diagram

i.e what would be an event for rolling a die?

2, 4, 6 ---collection of sample points

5 white balls drawn from 69 white ones and 1 red ball drawn from 26 red ones. order doesn't matter how many outcomes do we have

292,201,338 69cpr5 x 26cpr1 =

there are 5 books. how many different ways can these books be arranged?

5!

you have five books and are going to put three on a bookshelf. how many different ways can the books be ordered on the shelf?

5!/(5-3)! = 60 or do 5 perm. 3 on calculator to get 60

you have five books and are going to select three to read. how many different combinations are there, ignoring the order in which they are selected?

5!/3!(5-3)! = 10 ---or just do 5 count 3 on calculator = 10

the complement of an event A is denoted ___

A^c

use counting rule when order ______ matter

DOES NOT

how do we denote A intersection B

P(A,B)

how many outcomes for a password containing 8 characters (upper case, lower case, numbers)?

_ _ _ _ _ _ _ _ 26 upper case + 26 lower + 10 # = 62 8 slots so 62 ^ 8 = 2.18 x 10^14

marginal probability refers to the probability of ____. I.e P(jan)

a simple event

the union of A and B contains all sample points _____

belonging to A or B or both.

the intersection of A and B is the event containing the sample points belonging to

both A and B.

_____ an event that is sure to occur. what prob does this event have?

certain event; 1

which method to use when trying to find how many ways to get 2 heads when flipping 3 coins?

count function. k = 2 n = 3 3 count 2 = 3

give an example of an experiment and experimental outcomes of that experiment

experiment --> flipping coin outcomes --> heads or tails

give an example of independent events?

flipping a coin multiple times

two events are mutually exclusive if they ___

have no sample points in common. i.e A intersection B = 0

______ an event that has no chance of occurring; what probability does this event have?

impossible event; 0

3 types of counting rules

multiple step experiment _ _ _ _ permutations (order does matter) combinations (order doesn't matter)

joint probability refers to prob. of _______; i.e P(jan. and wed.)

occurrence of two or more events (joint event)

conditional probability is prob. of _____

one event, given that the other has occured

conceptually, if we were to shade the area of a Venn diagram of two circles intersecting, which section would represent conditional probability?

only the shades section where the two circles cross.

what does it mean if two events A and B are independent?

prob of B occurring wont change prob of A occurring

______ is the likelihood that an event will occur

probability

in probability what does an experiment mean?

process that generates well defined outcomes

____ the set of all experimental outcomes. An experimental outcome is also called _____

sample space. sample point

a day in January from all days in 2013 = _____ event a day in January that is also Wednesday from all days in 2013 = ____ event

simple joint

simple event vs. joint event

simple - describe by single characteristic joint - described by two or more characteristics

Describe the square and circle in the Venn diagram for visualizing probability

square = sample space (all experimental outcomes) circle = event (collection of experimental outcomes/sample points)

if we randomly select a day in 2013, A = day selected is jan, B = day selected is Wednesday; what is intersection of A and B

the day selected is Wednesday and in January

if we randomly select a day in 2013, A = day selected in January, B = the day selected is Wednesday; what is A union B mean

the day selected is in jan or is Wednesday

describe probability types on a contingency table. which sections of the table are joint and which are marginal.

the totals for each section as marginal; appear in the margins. joint appear in the body of the table

3 procedures for Bayes theorem

tree diagram then contingency table then posterior probabilities P(A given B) = P( A, B)/P(B)

an event is a collection of sample points t/f

true

when the problem asks specifically what is prob. ___ given ____, then use the conditional probability equation. when a problem gives already the probability of an event occurring given another event, it's going to want you to use the multiplication law equation.

true; practice slide 32

describe Bayes Theorem

we begin analysis with initial or prior probability. then from source, we obtain additional information about the events and update the probability values referred as posterior probability

are A and A^c mutually exclusive?

yes