Probability Flashcards
Probability summary,
can be expressed as
range
probability of zero
probability of 1
sum of all probabilities for an experiment
outcome
trial
sample space
event
Probabilities are expressed as fractions, decimals or percentages
Probabilities will always be within the range of 0 - 1 (0% - 100%)
If the probability is 0 the event cannot occur
If the probability is 1 the event is certain to occur
With all experiments the sum of the probabilities of all possible outcomes is 1
An outcome is the result of an experiment
A trial is each time the process of obtaining a result for an experiment is carried out
The sample space is the set of all possible outcomes
An event is part of the sample space
Theoretical probability,
description
formula
For equally likely outcomes, the theoretical probability of the event “E” occurring is given by:
P(E) =
Number of ways the event can occur / All possible outcomes (sample space)
Experimental probability
As well as working out the probabilities from theory, by performing an experiment and examining the results, the probability can be determined
Relevant frequency formula
Relevant frequency =
Number of times the event happened / Total trials (sample space)
Rule with Permutations and Combinations
Permutations, combinations and arrangements may be used to work out the probability
Venn Diagram description/uses
Venn diagrams are useful for showing sets and situations involving probability, they consist of a rectangle which shows the universal set (U) which means all possible outcomes (total sample space) and circle which show sets (events)
Venn diagram with two sets (events),
description
union diagram and formula
intersection diagram
disjoint diagram and formula
complement diagram, formula and note
Venn diagrams can contain two sets (events) and the following arrangements are some of those possible
Union of 2 sets (A U B)
A or B:
OO with overlap and all shaded
To find the probability of A U B add the probabilities of A and B. This will count the intersection twice so subtract the probability of the intersection
P(A U B) = P(A) + P(B) - P(A n B)
Intersection of 2 sets (A n B)
A and B
OO with overlap and only the overlap shaded
Disjoint A n B = ∅
O O with no overlap
To find the probability of A or B,
we add the individual probabilities of A and B,
P(A U B) = P(A) + P(B)
Complement of a set A’ (not A)
O with the box shaded around A
P(A) is the probability inside the circle
P(B) is the probability outside the circle
P(A) + P(B) = 1
If A is an event ,then A’ is the complement of A,
A’ means anything but A
Mutually exclusive events,
description
diagram
formula
notes
If an event A can occur OR an event B can occur, but not both A and B can occur then the events are said to be mutually exclusive
A and B both shaded with no overlap
P(A or B)
= P(A U B)
= P(A) + P(B)
This is sometimes known as the Addition Law for probabilities
For mutually exclusive events (A U B) = 0
Not mutually exclusive events,
description
diagram
formula
note
If an A and an event B can occur simultaneously they are not mutually exclusive
A and B both shaded with overlap
P(A or B)
= P(A U B)
= P(A) + P(B) - P(A n B)
It is essential to subtract the probability of A and B occuring together or this probability will be counted twice
Sketching a Venn diagram
- Put in the value of the intersection
- Deduct the value of the intersection from each remaining region for A and B
- The sum of all the probabilities assigned must be equal to 1
Tree diagram
A tree diagram or probability tree can help to solve probability or problems or problems involving the number of ways that a combination of things can be carried out , tree diagrams are useful for solving probability problems with more than one stage
Probability rules on a probability tree
- To find the probability of event A and event B happening multiply probabilities across the tree
- To find the probability of event A or event B happening add probabilities down the tree
Independent events,
description
formula
note
If either of the events A and B can occur without without being influenced by one another then the events are independent
P(A & B) = P(A n B) = P(A) x P(B)
Note:
This is sometimes known as the multiplication law for probabilities
Conditional probability,
description
formula x2
note
Conditional probability is the probability of an event A occurring given that some other event B has already occured
P(A|B) = [ P(A n B) ] / P(B)
or
P(A n B) = P(A|B) x P(B)
Notes:
The denominator is always the probability of the given even or the event that has already happened
If the events are independent then P(A|B) = P(A)
If the two events are mutually exclusive events then P(A|B) = 0 and they cannot both happen together
Application of conditional probability:
formula x3
P(A|B) = P(A n B) / P(B) –> P(A n B) = P(A|B) x P(B)
P(B|A) = P(B n A) / P(A) –> P(B n A) = P(B|A) x P(A)
P(A n B) = P(B n A) –> P(A|B) x P(B) = P(B|A) x P(A)
