If three cards are drawn in succession, how do we symbolize the events for drawing, from a regular deck of 52 playing cards, three specific cards in succession: A, B, and C? What formula would we use to determine the probability?
We draw three cards in succession from a regular deck of 52 playing cards. What is the probability that:• The first card is a red ace, AND• The second card is a 10 or a Jack, AND• The third card is greater than 3 but less than 7?
In a poker hand consisting of 5 cards, find the probability of holding 3 aces
In a poker hand consisting of 5 cards, find the probability of holding 4 hearts and 1 club
Set
any collection of objects
Union of A and B
The set of all elements in A or B or both
Intersection of A and B
The set of all elements in BOTH A and B
Complement of A
all values NOT in A
Event
Any subset of outcomes contained in the sample space S
Single Event
a single outcome
Compound event
multiple outcomes
Permutation
an ordered subset
Combination
An unordered subset
Cardinality of set |A|
Number of elements contained in A
Prime number
any whole number greater than one divisible only by one and itself
Interquartile Range
Range between Q1 and Q3
How to determine mild and major outliers
Mild: IQR*1.5Major: IQR*3.0
Random Variable (RV)
When is a random variable X discrete?
If the set of possible values is finite OR countable
When is a random variable X continuous
If the set of possible values of X is uncountable and X possesses a density function.
Two examples of continuous random variables
Real numbersAll the numbers in any interval
When are you not to use the function f(x) in probability?
In the case of a discrete random variable
Continuous Random Variable
Which of the following is a measure of probability?
f(x)
or
p(x)
What is the other?
p(x), the probability mass functionf(x) is a measure of frequency
What does F(3)-F(2) mean in words
The area under the distribution of probability between 3 and 2
How, in formula and words, does f(x) relate to F(x)
From a group of 5 women and 7 men, how many different committees consisting of 2 women and 3 men can be formed?
Suppose 5 components are installed in a series, and if any one of the components fails, the system is failed. How do we express this in terms of probability formulas?
Under what circumstances can A be a subset of B?
If every element in A is also in B
Under what circumstances are A and B considered to be equal?
If both A and B contain the same elements
If any operation can be performed in n_1 ways, and for each of these a second operation can be performed in n_2 ways, and for each of the first two a third operation can be performed and so on, up to a sequence of k operations, how many ways can the sequence of operations be performed?
Probability of an Event A, based on all sample points
If a probability can result in any of N different equally likely outcomes, and if exactly n of these outcomes correspond to event A, what is the probability of event A?
Additive rules for three events, A, B, and C
Subadditivity
Rule of Complements
Probability of an empty set
Property of Equally-Likely Outcomes
Conditional Probability/Independence/Product Rule
Suppose A and B are events in S. What are the conditions for the two events being independent?
x is an element of A
Not A/ the set of elements of S not in A
A is a subset of B
Cardinality of set A
Union of Two Sets A and B
Intersection of Two Sets A and
Sets of elements in A and not in B
A and B are disjoint or mutually exclusive
Integers are a subset of all real numbers
Natural Numbers are a subset of all integers
Cartesian products of A and B
Empty Set
Number of combinations of size r that can be formed from n objects in a set
Number of permutations of size r that can be formed from n objects in a set
Union from A1 to A3
Probability of A given B
P(A|B)
Is A={1,2,3,4,5,6} finitely countable, infinitely countable, or Uncountable?
Finitely countable?
Are all natural numbers finitely countable, infinitely countable, or uncountable?
Infinitely countable
If it is found that 210 out of 500 students smoke, what is the probability that a student in the sample smokes?
210/500 = 42%
Suppose that we know that 42% of students in a sample smoke, 51.6% of students drink, and 24.4% of students smoke and drink. What is the probability of a student randomly selected from the sample smoking but NOT drinking (P(m∩d'))
IF
P(m)=42%
P(e)=43.2%
P(e∩m)=19.4%
What is the probability of
P(e∪m)?
P(m∪e)=P(m)+P(e)-P(m∩e)=42+43.2-19.4=65.8%
If:
Probability component survives for more than 6000 hours is 0.42
Probability components survives for no longer than 4000 hours is 0.04
What is the probability that the life of the component is less than or equal to 6000 hours?
If:
Probability component survives for more than 6000 hours is 0.42
Probability components survives for no longer than 4000 hours is 0.04
What is the probability that the life of the component is greater than 4000 hours?
In sample of:
10 Juniors
30 Seniors
10 Graduates;
3 Juniors get A's
10 Seniors get A's
5 Graduates get A's
If a student has received an A, what is probability that they are a senior?
In a system, the whole system fails if component A, D, or either B or C fail
Probability of component surviving (independently):
P(A)=0.95
P(B)=0.7
P(C)=0.8
P(D)=0.9
What the probability that the system works?
P(E)=P(A)×P(C∪B)×P(D)=0.95×[P(B)+P(C)-P(B∩C)×0.9=0.95(0.7+0.8-0.7*0.8]×0.9=0.8037
Components A,B in series
Components C,D,E in series
AB in parallel with CDE
Probability of components surviving:
P(A)=0.7, P(B)=0.7, P(C)=0.8, P(D)=0.8, P(E)=0.8
What is probability that entire systems work?
P(A∪B)+P(C∪D∪E)-P(A∪B∪C∪D∪E)=0.7²+0.8³-(0.7²×0.8³)=0.75112
Departments A and B process a vaccine in sequential order independently of one another
Probability of Department rejecting vaccine:
P(A)=0.10
P(B)=0.08
What is the probability that a vaccine is passed by A but rejected by B?
P(A'∪B)=(1-P(A))×P(B)=0.9×0.08=0.072
P(A)=0.3, P(B|A)=0.75, P(B|A')=0.20, P(C|A∩B)=0.20
What is P(A∩B∩C)?
P(A)×P(B|A)×P(C|A∩B)=0.3×0.75×0.20=0.045
Probability of getting a full house in a five-card poker hand(note: Full house – 3 cards of same suit and 2 cards of same denomination)
How many denominations/types are there in an ordinary deck of playing cards
13
How many suits are there in an ordinary deck of playing cards
4
What is a joint probability distribution?
Symbol for “The set of possible values for X is 100 and 250”
What is the formula for the marginal p.d.f.’s of X and Y that can be obtained from a joint density function?
How do we determine the independence of two random variables, X and Y?
What is the range of a random variable?
The set of all possible values a random quantity could assume
Symbol for expected value of X
What is the formula for Discrete Mathematical Expectation?
What is Bayes' Rule?
What is the formula for Continuous Mathematical Expectation?
Also a random variable (measurable essentially means “well-behaved”, predictable)
What does it mean that a random variable is a constant?
What is variance a measure in a distribution?
The spread of the distribution
Because the expected value of a number is the number itself
What is covariance?
How two random variables vary together
Symbol for correlation coefficient of joint density function
What is the formula for finding the correlation coefficient of a joint density function? What is the interval this number is always in the interval of?
A perfect negative relationship
Uncorrelated (not necessarily independent)
What is the relationships between the correlation and whether X and Y are independent?
What is a linear combination of random variables?
