{3,5,6,7,9} (a) State A in standard roster form. A =. (b) Let B := A n{1,2,3,4,5, 6, 7,8}. State B in standard roster form. B = (c) Calculate v(p(B)). (d) Find p(B). P(B) :
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![Statement
Property
x= y
means "x is defined to be equal to y."
p V p = p pAp = p
pV q = q Vp p^q = q^p
p v (q v r) = (p v q) V r
p^ (q ^r) = (p ^ q) Ar
p V (q Ar) = (p V q) ^ (p V r)
p^ (q V r) ΦΛq) V φΛ γ)
pV (p ^ q) = p p^ (p V q) = p
pVl= p pAT = p
p V (¬p) = T p ^ (¬p) = 1
pVT =T pAl= 1
Idempotence
x:y
means "x is defined to be equivalent to y."
Commutativity
P = Q
means "P implies Q."
Associativity
means "P if and only if Q."
x ES
means "x is an element in the set S."
Distributivity
ACB
means "A is a subset of B."
Absorptivity
p:A → B
means "fis a function from the domain A to the codomain B."
Identity
$*(x)
means "the inverse function of f."
Complementarity
Dominance
(y • ¢)(x): = »($(x))
(Read "Þ following (or composed with) p.")
¬(¬p) = p
Involution
$>(A) := {$(a)|a e A}
(This is the "image" of A under ø.)
1= (T)느 T= (1)느
¬(p v q) = (¬p) ^ (¬q)
¬(p A q) = (-p) v (¬q)
Exclusivity
$<(B) := {a € A|p(a) € B}
(This is the "pre-image" of B under 4.)
DeMorgan's
Im(4) := {$(a)|a e Src(4)}
(This is the "image" of the function p.
Inference
Name
Inference
Name
Note: Im(4) = $>(Src($))).
(p) Cq)
Adjunction
Simplification
p
(p V q) (Gq)
Disjunctive
Syllogism
p
Addition
pv q
{ }
(the empty set)
p → (¬p)
Reductio Ad
Apagogical
Syllogism
N= {0,1,2,3,4, .}
(natural numbers)
-p
Absurdum
(p → q) (p)
(p → q) (¬a)
Z = {.., -3, –2,-1,0,1,2,3, .. }
(integers)
Modus Ponens
Modus Tollens
3{ (a,b e 2) A (b + 0)}
Q :=
(rational numbers)
(p → q) (q → r)
Hypothetical
Syllogism
Conditionalization
p →r
R:= {x |x is on the number line}
(real numbers)
(p → q) (r → )
(p V r) → (q V s)
(p → q) (p →r)
p → (q Ar).
(p → q) (¬p →r)
Resolvent
Dilemma
Complex Dilemma
qVr
(p → q) (¬p → q)
Exhaustive
Compositional
Syllogism
Syllogism
[a, b] := {x € R |a <x< b}
(Note: if a > b, [a, b] = { }and [a, a] = {a})
Fallacy
Name
Fallacy
Name
a..b = {n E Z ]asns b}
(Note if a > b, a..b = { }and a..a = {a})
(p → q) (q)
Asserting the
Conclusion
(p → q) (Gp)
Denying the
Premise
(p → ¬q) (q → -p)
(qר) p)-)A
(p → q) (p → r)
False
Elimination
Non-Sequitur
-(p A q)
False
Reduction
(p # q) (q ± r)
(p # r)
False Transition](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F004a8e3c-a278-4414-bbaf-77c8d726d7bd%2F9e3d4afb-df4f-4d07-9f21-e977a55e20d6%2Fh24ja8_processed.jpeg&w=3840&q=75)
Given
(b) To state in the roster form.
(c) To calculate .
(d) To find
Step by step
Solved in 3 steps
