BUS 19: Ch 4-Intro to Probability
The sum of the probabilities of a list of mutually exclusive and exhaustive events is
one
how many outcomes of an experiment constitute a simple event?
one
CONVERTING AN ODDS RATIO TO A PROBABILITY
Given odds for event A occurring of "a to b," the probability of A is a/a+b. Given odds against event A occurring of "a to b," the probability of A is b/a+b.
Calculate 5!
5*4*3*2*1= 120
Independent Events
Events for which the outcome of one event does not affect the probability of the other
CONVERTING A PROBABILITY TO AN ODDS RATIO
If P(A) denotes the probability of an event A occurring, and P(A) does not equal zero or one, then: The odds for A occuring equal P(A)/1−P(A), and The odds against A occuring equal 1−P(A)/P(A)
the probability of randomly selecting a "spade" from a deck of cards is___?
there are 52 cards in the deck, so the total number of outcomes is 52. But there are only 13 spades and 13 clubs, so that is the sample space. The probability of getting a spade, P(Spade), is13/52 or 0.2500.
Classical Probability
used in games of chance ~based on the assumption that all outcomes of an experiment are equally likely computed: the # of outcomes belonging to the event/the total number of outcomes
To calculate the probability of the union of two mutually exclusive events A & B.
we add the probability of A to the probability of B
complement rule
states that the probability of the complement of an event, P(Ac), is equal to one minus the probability of the event; that is, P(A^c)=1−P(A).
multiplication rule
states that the probability that A and Bboth occur is equal to the probability that A occurs given that B has occurred, times the probability that B occurs; that is,P(A∩B)=P(A | B)P(B). Equivalently, we can also arrive at this probability as P(A∩B)=P(B | A)P(A).
addition rule
states that the probability that A or B occurs, or that at least one of these events occurs, is equal to the probability that A occurs, plus the probability that B occurs, minus the probability that both A and B occur. Equivalently, P(A∪B)=P(A)+P(B)−P(A∩B).
A softball coach believes the Laurie has a 0.5 probability of getting a hit against a particular pitcher that Laurie has never batted against before. This type of probability is BEST characterized as a(n):
subjective probability
Conditional Probability
the probability that one event happens given that another event is already known to have happened
The probability that a customer orders popcorn at the movie theater is 0.40. The probability that a customer orders a drink at the movie theater is 0.65. The probability that a customer orders popcorn and a drink is 0.30. If a customer has already ordered a drink, what is the probability that the customer will order popcorn?
0.46
TWO DEFINING PROPERTIES OF PROBABILITY
1. The probability of any event A is a value between 0 and 1; that is, 0≤P(A)≤1. 2. The sum of the probabilities of any list of mutually exclusive and exhaustive events equals 1
Mike is placing a bet on an upcoming horse race in which seven horses are running. Mike places a trifecta bet that wins only if he correctly picks the first, second, and third place horses, in order. Mike can select the placing order for his bet in ___ ways.
210 n!/(n-x)! = 7!/(7-3)! = 5040/24 = 210
A festival has become so popular that it must limit the number of tickets it issues. People who hope to attend the festival send in a request for tickets, and request are filled by random selection. Only 21% of the ticket request are fulfilled. The odds that a random applicant does not receive a ticket are
3.76 to 1- The odds against A occurring equal 1-P(A)/P(A)
Dimitri is the coach of the high school athletes team. There are 8 mathletes, but only 5 may represent the school at the upcoming math tournament. Dimitri can choose 5 athletes from the 8 eligible athletes in ____ ways.
8!/(8-5)!*5!= 56
subjective probability
A probability value based on the individual's personal estimate and subjective judgement
THE ADDITION RULE FOR MUTUALLY EXCLUSIVE EVENTS
If A and B are mutually exclusive events, then P(A∩B)=0 and, therefore, the addition rule simplifies to P(A∪B)=P(A)+P(B).
The complement rule with respect to event A is
P(A^c) = 1 - P(A)
Which of the following is an example of a conditional probability?
The probability that Lisa purchases groceries, given that Neil has already purchased groceries
exhaustive
When all possible outcomes of an experiment are included in the event Example: one event represents "earning a medal" and the other denotes "failing to earn a medal." because they include all outcomes in the sample space. In the earlier grade-distribution example, the events of getting grades A and B are not exhaustive events because they do not include many feasible grades in the sample space. However, the events P and F, defined as "pass" and "fail," respectively, are exhaustive.
empirical probability
a probability value based on the observing the relative frequency with which an event occurs *must be repeated in large number of times for empirical probabilities to be accurate
experiment
a process that leads to one of several possible outcomes. The diversity of the outcomes of an experiment is due to the uncertainty of the real world.
The odds against a horse winning a race were set at 7 to 1. The probability of that horse not winning the race is
a/a+b+ 7/7+1= 0.875
The complement of event A within the sample space S contains
all outcomes in S that not in A
The __ formula is used to determine the number of different ways to arrange a group of x objects form a total of n objects when the order of the objects is irrelevant.
combination
sample space
denoted by S, of an experiment contains all possible outcomes of the experiment.
factorial formula
denoted n! i.e. given n items, there are n! ways of arranging them
the relative frequency of an event is used to calculate what type of probability?
empirical
mutually exclusive
events the do not share a common outcome in the event
events the cannot both occur on the same trial of an experiment are mutually ___ events
exclusive
Dependent Probability
if the occurrence of one is related to the probability of the occurrence of the other.
If two events do not influence each other, then the events are ___ events
independent
the probability that state college wins a football game is 0.60. The probability that
independent
complement
of event A, denoted A^c, is the event consisting of all outcomes in the sample space S that are not in A.
intersection
of two events, denoted A∩B, is the event consisting of all outcomes in A and B
union
of two events, denoted A∪B, is the event consisting of all outcomes in A or B
Probability
~is a numerical value that measures the likelihood that an event occurs. This value is between zero and one, where a value of zero indicates impossible events and a value of one indicates definite events.
event
~is a subset of the sample space. A simple event consists of just one of the possible outcomes of an experiment. ~Getting an A in a course is an example of a simple event. An event may also contain several outcomes of an experiment. ~For example, we can define an event as getting a passing grade in a course; this event is formed by the subset of outcomes A, B, C, and D.