FORMULAS TO MEMORIZE PROBABILITY Flashcards
some process that occurs with well defined outcomes
experiment
a result from a single trial of the experiment
outcome
a collection of one or more outcomes
event
a collection of all the outcomes of an experiment
sample space
“expected” probability based upon knowledge of the situation, such as the number of outcomes
theoretical probability
probability “estimate” based upon how often the event occured after collecting data from an experiment in a large number of trials
Empirical (experimental) probability
V = or/union
N= and/intersection
P(A or B)= P(A) + P(B) - P(A and B)
Probability Addition Rule
The two events have no outcomes in common (equal ZERO)
P (A and B) must be equal to 0
P (A or B) = P(A) + P(B)
Mutually Exclusive Events
Probability that event A occurs given that event B has already occured
“Probability of A given B”
P (A/B) = P(A and B)/P(B)
Conditional Probability
The probability that one event occurs in no way affects the probability of the other event occurring
ex. rolling a die and flipping a coin
Independent Events
The probability of one event occurring influences the likelihood of the other event
ex. Drawing a heart from a deck of cards, then drawing another heart from the deck
Dependent Events
Events A and B are independent if
P(A) = P(A/B)
OR
P(A) x P(B) = P (A and B)
Testing for Independence
*one may be easier over another depending on what information is offered
if an event has more than one stage to it, then a tree diagram can be drawn to list ALL the possible outcomes
Tree Diagrams/Sample Spaces
P (A THEN B) = P(A) x P(B)
Multiplication Rule for Multi Stage Events
If an object is replaced, and the probability of the second event does not change, then these events are independent
With replacement
If an object is not replaced, and the probability of the second event changes, the denominator will ALWAYS be one less, the numerator is sometimes changed as well, thus, these events are dependent
Without replacement
P(E) = n(E)/n(S)
n(E)= # of outcomes that fall into event E
n(S)= # of outcomes that fall into the sample space
Probability of an event E occurring
0 (impossible) < P(event) < 1 (certain event)
range of probability
P(Event) + P(NOT event) = 1
complement
visualizes and keeps track of all possible outcomes in a sample space (union, intersection)
Venn diagrams
P(neither) = 1 - P(at least one thing/policy)
chances of neither (opposite of or)
2, 3, 5, 7, 11, 13, 17, 19, 23
Prime numbers
the probability of an impossible event is 0 and the probability of a certain event 1
therefore the range of probability is 0 < P(E) <1
– –
P(E) = P(not E) = 1
(aka complement)
P(A or B)= P(A) + P(B) - P(A and B)
P(A V B) = P(A) + P(B) - P(A n B )
probability addition rule
P(A or B) = P(A) + P(B)
For mutually exclusive events (events with no outcomes in common)
P(A|B) = n(A and B)/n(B) or P(A and B)/p(B)
Conditional Probability
If the probability that one event occurs in no way affects the probability of the other event occurring
the events are independent
If the probability of one event occurring influences the likelihood of the other event
the events are dependent
Tests for Independence:
Events A and B are independent if:
1) P(A) = P(A|B)
2) P(A) x P(B) = P (A and B)
Multiplication Rule for multi-stage events P(A then B)= P(A) x P(B)
probability with replacement
if an object is replaced, the probability of the second event does not change
probability without replacement, if an object is not replaced, the probability of the second event will change (the denominator will always be one less and the numerator may be changed)
