PROBABILITY
COUNTING
If an event can occur in m ways and a second event can occur in n ways, the first event
followed by the second event can occur in mn ways, assuming the second event is no way
influenced by the first and this is the general principle of counting, sometimes called the
multiplication axiom.
The multiplication axiom can be extended to any number of events provided that no one
event influences another.
Example:
A teacher has five different books which he wishes to arrange on his desk. How many
different arrangements are possible?
Solution:
There are five choices ( event ) to be made, one for each space which will hold a book. To
select a book for the first space, the teacher has five choices. For the second space it has four
choices, for the third space has three choices and so on. By the multiplication axiom, we see
that the number of different arrangement is 5 x 4 x 3 x 2 x 1 = 120 ways. Therefore, there are
120 ways.
For convenience, the symbol n! ( read n factorial ) is used to denote such products. For any
counting number n, we define:
n! = n ( n-1 ) ( n-2 ) ( n- 3 )…….(2) (1)
By this definition of n! , we see that n ( n-1 )! = n! for all natural numbers n≥ 2. It is
convenient to have this relation hold also for n = 1, so we define; 0! = 1
PERMUTATION
If P(n,r) where r≤ n denotes the number of permutations of n elements taken r at a time,
then:
P ( n, r ) = (
𝑛!
𝑛−𝑟 )!
COMBINATION
If 𝐶(𝑛𝑟)
denotes the number of combinations of n elements taken r at a time, then:
𝐶(𝑛𝑟) =
𝑛!
( 𝑛−𝑟 )!𝑟!
PERMUTATION
COMBINATION
Number of ways of selecting r items out of n items
Repetitions are not allowed
Order is important
Order is not important
Arrangement of n items taken r at a time
Subsets of n items taken r at a time
P( n, r ) = (
𝑛!
C(𝑛𝑟) = (
𝑛−𝑟 )!
𝑛!
𝑛−𝑟 )!𝑟!
THEOREMS
1) If among n objects, r are alike and others are all distinct, the number of permutations of
these n objects taken all together is :
𝑛!
𝑟!
2) If among n objects r1 are identical, another r2 are identical and the rest are all distinct,
the number of permutations of these n objects taken all together is :
𝑛!
𝑟1!𝑟2!𝑟3!….
3) The number of ways in which n distinct objects can be arranged in a circle is : ( n-1 )!
PROBABILITY - the chance that a given event will occur ; the ratio of the number of outcomes
in an exhaustive sets of equally like outcomes that produce a given event to the total number of
possible outcomes.
Trial – repetition of an experiment.
Outcomes – possible results in each trial.
Example:
Tossing a fair coin
Two possible outcomes of each trial. One outcome is HEAD and the other is TAIL. If the coin
is not loaded to favour one outcomes over the other, it is called a fair coin. This means that the
two possible outcomes, H and T are equally likely or equiprobable. For a fair coin, this “ equally
likely “ assumption is made for each trial. Since there are two equally likely outcomes possible,
Head and tail, and just one of them is head, the probability of tossing a coin and getting Head is
1 divided by 2.
1
Probability( Head ) = P(H) = 2
Similarly;
1
Probability ( Tail ) = P(T) = 2
Sample Space – the set of all possible outcomes for an experiment.
S = {𝐻, 𝑇}
Event - a subset of a sample space.
E = {𝐻}
The Probability of an Event is defined as:
Probability ( event E ) = P ( E ) =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑖𝑛 𝐸
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑖𝑛 𝑆
where S is the sample space for event. In other words, the probability of an event is the
number of favourable outcomes divided by the total number of outcomes.
The set of all outcomes in the sample space that do not belong to an event E is called the
Complement of E, written E’. From its definition, we see that for any event E from a sample
space, we have:
E ∪ 𝐸′ = S
and
E ∩ 𝐸′ = 0
Odds – a comparison of P ( E ) with p ( E’ ).
The odds in favour of an event E is defined as the ratio of P ( E ) to P ( E’ ) or the fraction
𝑃(𝐸)
𝑃 ( 𝐸′ )
or vice – versa.
In general, if the odds favouring an event E are m to n, then :
𝑚
P ( E ) = 𝑚+𝑛
and P ( E’ ) =
𝑛
𝑚+𝑛
Tree Diagram – shows the various possible outcomes of an experiment. It contains the main
branches and spread as many branches it has depending on how many events there are.
Venn Diagram - one way in illustrating mutually or not mutually set of events. Events are
represented by circles and the universal set in rectangle as the whole sample space.
A
B
S
PROBABILITY OF AN ALTERNATE EVENTS
Two events which have no outcomes in common are said to be Mutually Exclusive Events. If
we use a Venn Diagram to illustrate mutually exclusive events, they will be disjoint. Three or
more events are mutually exclusive events if each pair of events is mutually exclusive.
The probability that event A or event B occurs is written as P ( A or B ) or in set symbol as
P ( A ∪ 𝐵). If A and B are mutually exclusive events, we can find P ( A ∪ B ) using addition rule
for mutually exclusive events:
P(A ∪ B) = P(A) + P(B)
For any experiment, events E and E’ are mutually exclusive. Therefore,
P ( E ∪ E’ ) = P ( E ) + P ( E’ )
However, E ∪ E’ = S , the entire sample space and P ( S ) = 1; so that P ( E ) + P ( E’ ) = 1
Therefore:
P ( E ) = 1 - P ( E’ )
and P ( E’ ) = 1 - P ( E )
For two events that are not mutually exclusive, use the following property:
P( A ∪ B) = P(A) + P(B) - P( A ∩ B )
In words, the probability that event A or B happens, is the probability that event A happens plus
the probability that event B happens minus the probability that both events happen.