PROBABILITY AND COUNTING TECHNIQUES
“KEY CONCEPTS AND FORMULAS”
(A) SETS
1. DEFINITION OF A SET: A set is a welldefined collection of distinct objects, called the elements or
members of the set. One way of describing a set is by listing its elements in a “Roster.”
For example: A = {a, e, i, o, u}; X = {0, 8, 2, 3, 7, 10}; etc.
NOTE: When the list is extensive and the pattern is obvious, an ellipsis may be used.
For example: B = {a, b, c, …, z}.
NOTE: The above form is also called the Tabular Form of the Set. The elements of a set are separated by commas and are enclosed in brackets { }.
2. SUBSET: A is a subset of B if every element in set A is also in set B. The notation is A Í B. NOTE:
Every set is a subset of itself, i.e., A Í A, where A is any set.
3. NULL OR EMPTY SET: The null or empty set is the set with no elements. The notation is Æ or { }.
NOTE: The empty set is a subset of every set, i.e., Æ Í A, where A is any set.
4. UNIVERSAL SET: The UNIVERSAL SET, U, is the set that contains all possible elements under consideration in a problem.
5. NUMBER OF SUBSETS OF A SET: If a set contains “N” elements then the number of subsets is equal to 2^{N} and the number of proper subsets is equal to 2^{N}  1.
6. POWER SET: The set of all the distinct subsets of a set A is called the power set of A. The notation is P(A). Thus, if n(A) = N, then n(P(A)) = 2^{N}, where A is any finite set.
7. UNION OF SETS: The union of sets A and B is the set of all elements which are either in A or in B. The notation is A È B. Thus A È B = {x  x Î A or x Î B or x Î both A and B}.
8. INTERSECTION OF SETS: The intersection of sets A and B is the set of all those elements, which are common to both A and B. The notation is A Ç B. Thus A Ç B = {x  x Î A and x Î B}.
9. COMPLEMENT OF A SET: Let A be a set and U the universal set. Then the complement of set A with respect to the universal set U is the difference of sets U and A, i.e., U – A, and is denoted by
_
A^{/ }or A. Thus A^{/} = {x  x Î U and x Ï A}.
10. DISJOINT SETS: Any two sets A and B are said to be disjoint if and only if A Ç B = Æ.
11. VENN DIAGRAMS: We can visualize sets and operations on sets and guess the truth of a number of propositions on sets with the help of geometric diagrams, known as Venn diagrams. A convenient way to use Venn diagrams is to represent the universal set U by rectangular area in a plane and the elements which make up U by the points of this area. Simple plane areas bounded by circles or their parts drawn within the rectangular area can give a simple and instructive picture of sets and operations on sets. We can think of each set as consisting of all points within the corresponding circle.
(B) PROBABILITY
1. PROBABILITY EXPERIMENT: A probability experiment refers to any act or process or procedure that can be performed that yields a collection of outcomes (or results). The outcome of the probability experiment is not known in advance of the act. For example:
(i) The act of tossing a fair coin.
(ii) The act of rolling a fair die.
(iii) The act of drawing a single card from an ordinary deck of 52 playing cards.
2. SAMPLE SPACE: A sample space is defined as the set of all possible outcomes (or results) that can occur in a probability experiment such that exactly one outcome occurs at a time. The letter S will be used to denote a sample space. In a Venn diagram, a rectangle will be used to represent a sample space and the outcomes which make up S by the points drawn within this rectangle.
For example:
(i) If a fair coin is tossed, there are two possible, equally likely outcomes in the sample space that could occur in this experiment (Heads, H, or Tails, T). Thus S = {H, T}.
(ii) If a fair die is rolled, then S = {1, 2, 3, 4, 5, 6}.
3. EVENT: An event is any collection of results or outcomes of a probability experiment. Thus, an event may be defined as any subset of the sample space. For example:
(i) If a fair coin is tossed, each of the two individual outcomes (Heads, H, or Tails, T) may be referred to as events since the sets {H} and {T} are subsets of S = {H, T}.
(ii) If a fair die is rolled, then each of the individual outcomes in this probability experiment is an event because {1}, {2}, {3}, {4}, {5}, and {6} are subsets of S = {1, 2, 3, 4, 5, 6}.
Similarly, {1, 3, 5} and {2, 4, 6} represent the events of rolling an odd number and an even number on the die respectively. Note that no two outcomes can occur at the same time.
The event of rolling a 7 on a die is Æ or { } because Æ is a subset of S, i.e. Æ Í S.
Notations: The capital letters such as A, B, C, D, E, etc. will be used to denote an event. In a Venn diagram, a circle drawn within the rectangle will be used to represent an event and the outcomes which make up the event by the points of this circle.
4. SIMPLE EVENT: A simple event is an outcome or an event that cannot be further broken down. For example:
(i) If a fair coin is tossed, each of the two individual outcomes (Heads, H, or Tails, T) may be referred to as simple events.
(ii) If a fair die is rolled, then each of the individual outcomes {1}, {2}, {3}, {4}, {5}, and {6} is a simple event.
5. COMPOUND EVENT: A compound event is any event combining two or more simple events. For example:
(i) If a fair die is rolled, then {1, 3, 5} and {2, 4, 6} may be referred to as compound events.
6. COMPLEMENT OF AN EVENT: Let E be any event and S the sample space of a probability experiment. Then the complement
_
of the event E with respect to the sample space S, denoted by E^{/ }or E, consists of those outcomes which are not in E, i.e.
E^{/} = {x  x Î S and x Ï E}. For example:
(i) If a fair die is rolled, then {1, 3, 5} and {2, 4, 6} may be referred to as complementary events of each other.
7. FORMULAS:
Let_{} and _{} be any two events from the sample space _{}of a probability experiment. Let _{} the number of outcomes in _{} and _{} the number of outcomes in_{}. Let _{}denote the probability of event _{} occurring, and _{} denote the probability of event _{} occurring.
(i) PROBABILITY OF AN EVENT: _{} (Classical Probability)
(ii) PROBABILITY OF AN IMPOSSIBLE EVENT: If the event _{} cannot occur, then_{}.
(iii) PROBABILITY OF A CERTAIN EVENT: If the event _{} is certain to occur, then_{}.
(iv) The probability of any event _{} is a number between _{} and _{}, inclusive, i.e. _{}
(v) ADDITION RULES:
(a) FOR MUTUALLY EXLUSIVE EVENTS: The events _{} and _{} are called mutually exclusive (or disjoint) if _{}, i.e. they cannot both occur together, i.e. simultaneously. For mutually exclusive events, we have _{}, and the addition rule is defined as follows:
_{}.
(b) FOR NONMUTUALLY EXLUSIVE EVENTS: The events _{} and _{} are not mutually exclusive if _{}, i.e. they both occur simultaneously. In this case, the addition rule is defined as follows:
_{}.
(vi) MULTIPLICATION RULES:
(a) FOR INDEPENDENT EVENTS: The events _{} and _{} are called independent if the occurrence of one event, say, _{} does not affect the probability of the occurrence of the other event_{}. The multiplication rule is defined as follows:
_{}_{}.
(b) FOR DEPENDENT EVENTS: The events _{} and _{} are called dependent if the occurrence of one event, say, _{} affects the probability of the occurrence of the other event_{}. In this case, the multiplication rule is defined as follows:
_{},
where _{} is read as “_{}” and _{}represents the probability of event _{} occurring after it is assumed that event _{} has already occurred.
8. FORMULAS (CONTINUED):
Let_{} and _{} be any two events from the sample space _{}of a probability experiment. Let _{} the number of outcomes in _{} and _{} the number of outcomes in_{}. Let _{}denote the probability of event _{} occurring, and _{} denote the probability of event _{} occurring.
(i) CONDITIONAL PROBABILITY RULE:
(a) DEFINITION: A conditional probability of an event _{} represents the probability of event _{} occurring after it is assumed that some other event _{} has already occurred. It is denoted by _{}.
(b) FORMULA: The conditional probability of event _{} occurring, given that event _{} has already occurred, is denoted by _{}, and is given by
_{} , provided _{}, or _{}.
(ii) PROBABILITY OF THE COMPLEMENT OF AN EVENT: Let _{} denote the complement of the event _{} with respect to the sample space _{}. Then
_{}The probability that the event _{} will not occur _{};
_{}_{}; and _{}_{}.
(iii PROBABILITY OF “AT LEAST ONE”:
(a) “At least one” is equivalent to “one or more.”
(b) “None” and “at least one” are complement of each other.
(c) P(at least one) = 1 – P(none) and P(none) = 1 – P(at least one)
(iv) TESTING FOR INDEPENDENCE:
Two events _{} and _{} are “independent” if 
Two events _{} and _{} are “dependent” if 
_{}
OR
_{}

_{}
OR
_{} 
(v) BAYES’ FORMULA:
_{}
9. FORMULAS FOR ODDS:
Let_{} be any event from the sample space _{}of a probability experiment. Let _{} the number (#) for (in favor of) the event _{}, and _{} the number (#) against the event _{}. Let _{} = total number (#) of outcomes in _{}. Then, we have the following definitions:
(a) Odds in favor (for) of the event _{} _{}, or _{}, i.e.
Odds in favor (for) of the event _{} _{};
(b) Odds against the event _{} _{}, or _{}, i.e.
Odds against the event _{} _{};
(c) _{} _{}, which is the probability in favor of the event _{};
(d) _{} _{}, which represents the probability against the event _{}.
(C) COUNTING TECHNIQUES
(i) FUNDAMENTAL PRINCIPLE OF COUNTING:
For a sequence of two events (or choices or tasks), say, _{} and_{}, in which the first event _{} can occur _{} ways and the second event _{} can occur _{} ways, the events _{} and_{} together can occur a total of _{} ways. Here, none of the two events (or choices or tasks) depends on another.
NOTE: The fundamental counting principle easily extends to situations involving more than two events.
(ii) FACTORIAL NOTATION:
If _{} denotes an integer, then the factorial symbol _{} is defined as follows:
_{};
_{};
_{}
(iii) FACTORIAL RULE:
A collection of _{} different items (or objects) can be arranged in order _{} different ways.
(iv) PERMUTATION RULE (WHEN ITEMS ARE ALL DIFFERENT):
(a) DEFINITION: A permutation is a sequential arrangement of _{} different (or distinct) items (or objects) taken _{} at a time, without replacement or repetition, denoted by _{}, where _{}, in which the order makes a difference.
(b) FORMULA: _{} , where _{}.
(c) SHORTCUT FORMULA: _{} , where _{}.
(d) NOTE: _{}
(v) COBINATION RULE:
(a) DEFINITION: A combination is a group (or set or collection or selection) of _{} different (or distinct) items (or objects) taken _{} at a time, denoted by _{} or _{}, where _{}, in which the order is not important.
(b) FORMULA: _{} , where _{}.
(c) NOTE: _{}
(vi) PERMUTATION RULE (WHEN SOME ITEMS ARE IDENTICAL OR SIMILAR OR ALIKE TO OTHERS):
If there are _{} items (or objects) with _{} alike, _{} alike, … , _{} alike, the number of permutations of all _{} items (or objects) is given by
_{} , where _{}.
(vii) APPLICATIONS OF PERMUTATION AND COMBINATION RULES TO PROBABILITY:
(a) The probability of a permutation (arrangement) is given by
_{}
(b) The probability of a selection (combination) is given by
_{}
(viii) AN USEFUL FORMULA ON SIMPLE RANDOM SAMPLING:
(a) DEFINITION OF A SIMPLE RANDOM SAMPLE:
A sample of size _{} from a population of size _{} is obtained through simple random sampling if every possible sample of size _{} has an equally likely chance of occurring. The sample is then called a simple random sample.
NOTE: In the above definition, the sample is always a subset of the population with _{}.
(b) FORMULA: The number of different simple random samples of size _{} from a population of size _{} is defined as a combination of _{} objects by selecting _{} at a time without replacement, and is given by the following formula:
_{} , where _{}.