PROBABILITY Flashcards
branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true.
PROBABILITY
base formula for PROBABILITY
P= NUMBER OF DESIRABLE OUTCOMES / NUMBER OF ALL POSSIBLE OUTCOMES.
1st princicple of counting;
IN PROBABILITY WHEN AN EVENT CAN BE DONE IN M DIFFERENT WAYS AND ANOTHER EVENT CAN BE DONE IN N DIFFERENT WAYS, THEN THESE TWO EVENTS ‘CAN’ BE DONE ONE AFTER THE OTHER.
how does this two event relate through mathematical expression
M X N
- THE WORD “AND” IS USUALLY THE INDICATOR FOR MULTIPLICATION
- becuase all events will happen
2nd princicple of counting;
IN PROBABILITY WHEN AN EVENT CAN BE DONE IN M DIFFERENT WAYS AND ANOTHER EVENT CAN BE DONE IN N DIFFERENT WAYS, THEN THESE TWO EVENTS ‘CAN’ BE DONE IN TWO SEPARATE WAYS
how does this two event relate through mathematical expression
M + N
- THE WORD “or” IS USUALLY THE INDICATOR FOR addition
- only 1 event will occur
- SUBTRACTION IS ALSO USED IF THERE ARE REPEATING EVENT!!!
an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements.
permutation
-order is important! meaning arrangement 1-2 is different as 2-1
“combination lock should be called permutation lock because 123 is NOT the same as 213 in a real life combination lock”
formulae for PERMUTATION and explain each
nPr = n!/(n-r)! =when n objects is arranged at r at a time nPn= n! =when n objest is arranged at n time as well
THE NUMBER OF POSSIBLE OUTCOME = ____+ ____?
UNDSIRABLE OUTCOME + DESIRABLE OUTCOME!
the shifting of an entire of elements one or more stapes forward or backwards without without changing the order of the elements in the sequence.
CYCLIC PERMUTATION
formulas for cyclic permutation and explain each
nPr/r =when n objects is arranged at r at a time
nPn/n= (n-1)! =when n objest is arranged at n time as well
the number of permutation _____ when a collection contains identical elements
are REDUCED!
LIKE word arrangements, because M I S S is still M I SS when you switch the letter S, right?!?
GIVE THE FORMULA WHERE THE PERMUTATION OS N OBJECTS TAKEN ALL AT ONCE WITH P,Q,&S THINGS ALIKE…
LIKE WORD ARRANGEMENT WITH REPEATING LETTERs
n!/p!q!s!
is a selection of all or part of a set of objects, without regard to the order in which objects are selected.
combination
formula of Conbination
n!/r!(n-r)!
two or more events that can never happen in the same trial.
MUTUALLY EXCLUSIVE
THIS IS ONLY APPLICABLE FOR ADDITION RULE! (OR)
WHICH STATES
Pr + Pb = (P r or b)= P(rub)
two or more events that can happen in the same trial.
non mutually exclusive
also applies to ADDITION RULE! (OR)
with the formula
P(AorB)= P(AUB)= P(A)+P(B)-P(AnB)
IF THE OUTCOME OF ONE TRIAL DOES NOT AFFECT THE OUTCOME OF OTHER TRIALS THEN IT IS CALLED _____
INDEPENDENT EVENTS
muliplication rule is applied
P(AnB) = P(A) X P(B)
IF THE OUTCOME OF ONE TRIAL “AFFECTS” THE OUTCOME OF OTHER TRIALS THEN IT IS CALLED _____
DEPENDENT EVENTS
also applies mltiplication rule
given the formula;
P(B|A)= P(B GIVEN A)
P(AnB)= P(A) X P(B|A)
A MEASURE OF THE PROBABILITY OF AN EVENT OCCURING GIVEN THAT ANOTHER EVENT HAS OCCURED
CONDITIONAL PROBABILITY
-INTRODUCED BY THOMAS BAYES
P(A|B) = P(AnB)/P(B)
A CONDITIONAL RULE SAYS THAT, THE PROBABILITY OF A GIVEN B MULTIPLIED BY THE PROBABILITY OF OF B IS EQUAL TO ?
PROBABILITY OF B GIVEN A MULTIPLIED BY THE PROBABILITY OF A.
A “FREQUENCY DISTRIBUTION” OF THE POSSIBLE “NUMBER OF SUCCESSFUL OUTCOMES” IN A GIVEN NUMBER OF TRIALS IN EACH OF WHICH THERE IS THE “SAME PROBABILITY OF SUCCESS”
BINOMIAL DISTRIBUTION
-APPLICABLE FOR REPEATED TRIALS
4 CHARACTERISTICS OF A BINOMIAL DISTRIBUTION.
- an event is repeated in a fix number of trials
- only two possible outcomes “bi”
- the probability of success is same in each trial
- each trial is independent of each trial
BINOMIAL DISTRIBUTION FORMULA
P = (nCr )(p^r) q^(n-r)
P = binomial probability r = number of successful trials p = probability of success on a single trial q = probability of failure on a single trial n = number of trials
*where p + q = 1
a DISCRETE PROBABILITY DISTRIBUTION OF THE NUMBER OF EVENTS OCCURING IN A GIVEN TIME PERIOD, GIVEN THE AVERAGE NUMBER OF TIMES THE EVENT OCCURS OVER THAT TIME PERIOD
POISSON DISTRIBUTION.
– SIMILAR TO BINOMIAL DIST, BUT WITH A GIVEN TIME PERIOD.
4 CHARACTERISTICS OF POISSON DISTRIBUTION
- an event can occur any number of times in during
a time period - event occur independently
- the rate of occurence is constant
- the probability of an event occuring is proportional to the length of the time period
POISSON FORMULA IS ????? STATE THE FORMULA!!!
P(x=k) = (λ^k) (e^-λ) / k!
k= number of (events observed/successful trial) in a given time period. λ= expected value or (average/mean) of X
λ=np = MEAN
where, n= total number of trial & p= probability of success
σ = npq = VARIANCE
where q= probability of failure.
IN ORDER TO APPLY POISSON DISTRIBUTION FORMULA, THE VAULE OF λ MUST BE?
λ<10
A PROBABILITY FUNCTION THAT DESCRIBES HOW THE VALUES ARE DISTRIBUTED. A SYMMETRIC DISTRIBUTION WHERE MOST OF THE OBSERVATIONS CLUSTERS AT THE CENTRAL PEAK
NORMAL DISTRIBUTION
MEAN=MODE=MEDIAN
DIFFERENTIATE MEAN MEDIAN AND MODE
MEAN= AVERAGE
MEDIAN= THE MIDDLE VALUE FROM THE SET.
MODE= NUMBER THAT REPEATS THE MOST
RANGE=HIGHEST - LOWEST VALUE IN A SET.
4 PROPERTIES OF NORMAL DISTRIBUTION
- SYMMETRIC AT THE CENTER
- MEAN MEDIAN MODE IS EQUAL
- EX ATCLY HALF OF THE VALUE IS IN THE LEFT & RIGHT OF THE CENTER
THE TOTAL AREA UNDER THE CURVE IS 1-
FOR NORMAL DISTRIBUTION, THIS GIVES THE IDEA OF HOW FAR FROM THE MEAN A DATA POINT IS.
Z- SCORE, STANDARD SCORE
FORMULA FOR Z-SCORE
z=x-µ/σ
µ= MEAN
x=GIVEN VALUE
σ=standard dev
for normal distribution, formula for area to the left of (z =a) .probability where (a>z)
P(a>z)= ∫(-∞,a) [1/√(2π) e^(-z^2/2) dz
P(az)
GIVEN VALUES TO USE;
BINOMIAL DISTRIBUTION-
POISSON’S DISTRIBUTION-
NORMAL DISTIBUTION-`
BINOMIAL DISTRIBUTION- PROBABILITY OF ONE SUCCESSFUL TRIAL
POISSON’S DISTRIBUTION- EXPECTED VALUE
NORMAL DISTIBUTION-`MEAN & STANDARD DEV.
_____ IS A FORMULA FOR DETERMINING CONDITIONAL PROBABILITY THAT ALLOWS US TO UPDATE PREDICTED PROBABILITY OF AN EVENT BY ADDING IN NEW EVIDENCES.
BAYES THEOREM
BAYES THEOREM FORMULA
P(H|E)=[P(E|H)P(H)] / P(E)`
P(h) = previous likelihood P(e|H) = probability you want to know P(E) = total probability
probabillity of having at least 1/2/3 ….
a helpful equation can be ___
1 - ( opposite probability, e.g none where picked)
