COUNTING RULES
Statistics and ProbabilityCOUNTING RULES
- Permutations
- Combinations
- Random Experiment
- Sample Space
- Events
- Mutually Exclusive Events
- Exhaustive Events
- Equally Likely Events
- Events
COUNTING RULES
As discussed in the last lecture, there are certain rules that facilitate the calculations of probabilities in certain situations. They are known as counting rules and include concepts of;
- Multiple Choice
- Permutations
• Combinations We have already discussed the rule of multiplication in the last lecture. Let us now consider the rule of permutations.
RULE OF PERMUTATION
A permutation is any ordered subset from a set of n distinct objects. For example, if we have the set {a, b}, then one permutation is ab, and the other permutation is ba. The number of
n
permutations of r objects, selected in a definite order from n distinct objects is denoted by the symbol P and is given
r
by
n
P =n (n– 1)(n –2)…(n– r+ 1)
r
n!
=
_{) }^{.}
(n − r!
FACTORIALS
7! =7 × 6 × 5 × 4 × 3 × 2 × 1 6! =6 × 5 × 4 × 3 × 2 × 1 .
. . 1! =1
Also, we define 0! = 1.
EXAMPLE
A club consists of four members. How many ways are there of selecting three officers: president, secretary and treasurer? It is evident that the order, in which 3 officers are to be chosen, is of significance. Thus there are 4 choices for the first office, 3 choices for the second office, and 2 choices for the third office. Hence the total number of ways in which the three offices can be filled is 4 × 3 × 2 = 24. The same result is obtained by applying the rule of permutations:
4!
4
P_{3 }=
(4 − 3)! = 4× 3× 2 = 24
Let the four members be, A, B, C and D. Then a tree diagram which provides an organized way of listing the possible arrangements, for this example, is given below:
President Secretary Treasurer C B
D B C D B
D C C
A D A
C D A
D C B
A D A
B D A
D B B
A C AB C AC B
PERMUTATIONS
n
In the formula of P , if we put r = n, we obtain:
r n
P =n (n– 1)(n –2)…3 × 2 × 1
n
= n!
I.e. the total number of permutations of n distinct objects, taking all n at a time, is equal to n!
EXAMPLE
Suppose that there are three persons A, B & D, and that they wish to have a photograph taken. The total number of ways in which they can be seated on three chairs (placed side by side) is
3P3= 3! =6
These are: ABD, ADB, BAD, BDA, DAB, DBA The above discussion pertained to the case when all the objects under consideration are distinct objects. If some of the objects are not distinct, the formula of permutations modifies as given below: The number of permutations of n objects, selected all at a time, when n objects consist of n1 of one kind, n2 of a second kind, …, nk of a kth kind,
n!
is P = .
n !n !…..n !
12 k
(where ∑ n = n)
i
EXAMPLE
How many different (meaningless) words can be formed from the word ‘committee’? In this example: n = 9 (because the total number of letters in this word is 9) n1 = 1 (because there is one c) n2 = 1 (because there is one o) n3 = 2 (because there are two m’s) n4 = 1 (because there is one i) n5 = 2 (because there are two t’s) and n6 = 2 (because there are two e’s) Hence, the total number of (meaningless) words
(permutations) is: | |||||||||||||
!n | |||||||||||||
P | = | . | |||||||||||
1n | ! | 2n | ! ….. | kn | ! | ||||||||
9! | |||||||||||||
= | |||||||||||||
1! 1! | 2! 1! | 2! | 2! | ||||||||||
3456789 ××××××× | 12 × | ||||||||||||
=
1× 1× 2 × 1× 1× 2 × 1× 2 × 1 = 45360 Next, let us consider the rule of combinations.
RULE OF COMBINATION
A combination is any subset of r objects, selected without regard to their order, from a set of n distinct objects. The total number of such combinations is denoted by the symbol
_{n } n
C or
r
and is given by _{r}
n n!
^{= }
_{⎝}r _{ }r!(n − r)!
where r<n. It should be noted that
n
n
P= r!
r
r
In other words, every combination of r objects (out of n objects) generates r! Permutations
EXAMPLE
Suppose we have a group of three persons, A, B, & C. If we wish to select a group of two persons out of these three, the three possible groups are {A, B}, {A, C} and {B, C}.In other words, the total number of combinations of size two out of this set of size three is 3.
Now, suppose that our interest lies in forming a committee of two persons, one of whom is to be the president and the other the secretary of a club. The six possible committees are:
(A, B), (B, A), (A, C), (C, A), (B, C)& (C,B)
In other words, the total number of permutations of two persons out of three is 6. And the point to note is that each of three combinations mentioned earlier generates 2 = 2! Permutations, I.e. the combination {A, B} generates the permutations (A, B) and (B, A) and the combination {A, C} generates the permutations (A, C) and (C, A); and the combination {B, C} generates the permutations (B, C) and (C, B).
The quantity
n
r
n
or C is also called a binomial coefficient because of its appearance in the binomial expansion of
r
n
nn−rr
(a + b) =_{∑ }^{n} ab .
r =0 r
The binomial coefficient has two important properties.
n n
i) _{⎟}=, and
rn − r
n n n +1
ii)_{⎜⎟}=
^{+ }
n − rr r
Also, it should be noted that
n n
1 =
_{0}^{= }_{n }
and
n n
n =
1 n −1
EXAMPLE
A threeperson committee is to be formed out of a group of ten persons. In how many ways can this be done? Since the order in which the three persons of the committee are chosen, is unimportant, it is therefore an example of a problem involving combinations. Thus the desired number of combinations is
n 10 10! 10!
_{⎜}_{⎟}= =^{= }
_{ }r _{⎠ }3 _{ }3! (10 − 3)! 3! 7!
10× 9×8× 7 × 6× 5× 4× 3× 2×1
=
3× 2×1× 7 × 6× 5× 4× 3× 2×1 = 120
In other words, there are one hundred and twenty different ways of forming a threeperson committee out of a group of only ten persons!
EXAMPLE
In how many ways can a person draw a hand of 5 cards from a wellshuffled ordinary deck of 52 cards? The total number of ways of doing so is given by
n52 52 × 51× 50 × 49 × 48
= 2,598,960
^{=}^{= }
r 55× 4 × 3× 2 ×1
Having reviewed the counting rules that facilitate calculations of probabilities in a number of problems, let us now begin the discussion of concepts that lead to the formal definitions of probability. The first concept in this regard is the concept of Random Experiment. The term experiment means a planned activity or process whose results yield a set of data. A single performance of an experiment is called a trial. The result obtained from an experiment or a trial is called an outcome.
RANDOM EXPERIMENT
An experiment which produces different results even though it is repeated a large number of times under essentially similar conditions is called a Random Experiment.
The tossing of a fair coin, the throwing of a balanced die, drawing of a card from a wellshuffled deck of 52 playing cards, selecting a sample, etc. are examples of random experiments.
PROPERTIES OF A RANDOM EXPERIMENT
A random experiment has three properties:
- The experiment can be repeated, practically or theoretically, any number of times.
- The experiment always has two or more possible outcomes. An experiment that has only one possible outcome is not a random experiment.
• The outcome of each repetition is unpredictable, i.e. it has some degree of uncertainty. Considering a more realistic example, interviewing a person to find out whether or not he or she is a smoker is an example of a random experiment. This is so because this example fulfils all the three properties that have just been discussed:
- This process of interviewing can be repeated a large number of times.
- To each interview, there are at least two possible replies: ‘I am a smoker’ and ‘I am not a smoker’.
- For any interview, the answer is not known in advance i.e. there is an element of uncertainty regarding the
person’s reply. A concept that is closely related with the concept of a random experiment is the concept of the Sample Space.
SAMPLE SPACE
A set consisting of all possible outcomes that can result from a random experiment (real or conceptual), can be defined as the sample space for the experiment and is denoted by the letter S. Each possible outcome is a member of the sample space, and is called a sample point in that space. Let us consider a few examples:
EXAMPLE1
The experiment of tossing a coin results in either of the two possible outcomes: a head (H) or a tail (T). (We assume that it is not possible for the coin to land on its edge or to roll away). The sample space for this experiment may be expressed in set notation as S = {H, T}. ‘H’ and ‘T’ are the two sample points.
EXAMPLE2
The sample space for tossing two coins once (or tossing a coin twice) will contain four possible outcomes denoted by S = {HH, HT, TH, TT}. In this example, clearly, S is the Cartesian product A × A, where A = {H, T}.
EXAMPLE3
The sample space S for the random experiment of throwing two sixsided dice can be described by the Cartesian product A × A, where A = {1, 2, 3, 4, 5,6}. In other words, S = A × A ={(x,y)|x ∈ A and y ∈ A}, Where x denotes the number of dots on the upper face of the first die, and y denotes the number of dots on the upper face of the second die. Hence, S contains 36 outcomes or sample points, as shown below:
S = {(1, 1), (1, 2), (1, 3), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 5), (6, 6)}
The next concept is that of events.
EVENTS
Any subset of a sample space S of a random experiment, is called an event. In other words, an event is an individual outcome or any number of outcomes (sample points) of a random experiment.
SIMPLE & COMPOUND EVENTS
An event that contains exactly one sample point, is defined as a simple event. A compound event contains more than one sample point, and is produced by the union of simple events.
EXAMPLE
The occurrence of a 6 when a die is thrown, is a simple event, while the occurrence of a sum of 10 with a pair of dice, is a compound event, as it can be decomposed into three simple events (4, 6), (5, 5) and (6, 4).
OCCURRENCE OF AN EVENT
An event A is said to occur if and only if the outcome of the experiment corresponds to some element of A.
EXAMPLE
Suppose we toss a die, and we are interested in the occurrence of an even number. If ANY of the three numbers ‘2’, ‘4’ or ‘6’ occurs, we say that the event of our interest has occurred. In this example, the event A is represented by the set {2, 4, 6}, and if the outcome ‘2’ occurs, then, since this outcome is corresponding to the first element of the set A, therefore, we say that A has occurred.
COMPLEMENTARY EVENT
The event “notA” is denoted by A or Ac and called the negation (or complementary event) of A.
EXAMPLE
If we toss a coin once, then the complement of “heads” is “tails”. If we toss a coin four times, then the complement of “at least one head” is “no heads”. A sample space consisting of n sample points can produce 2n different subsets (or simple and compound events).
EXAMPLE
Consider a sample space S containing 3 sample points, i.e. S = {a, b, c}. Then the 23 = 8 possible subsets are
φ, {a}, {b}, {c}, {a, b},
{a, c}, {b, c}, {a, b, c} Each of these subsets is an event. The subset {a, b, c} is the sample space itself and is also an event. It always occurs and is known as the certain or sure event. The empty set φ is also an event, sometimes known as impossible event, because it can never occur.
MUTUALLY EXCLUSIVE EVENTS
Two events A and B of a single experiment are said to be mutually exclusive or disjoint if and only if they cannot both occur at the same time i.e. they have no points in common.
EXAMPLE1
When we toss a coin, we get either a head or a tail, but not both at the same time. The two events head and tail are therefore mutually exclusive.
EXAMPLE2
When a die is rolled, the events ‘even number’ and ‘odd number’ are mutually exclusive as we can get either an even number or an odd number in one throw, not both at the same time. Similarly, a student either qualifies or fails, a single birth must be either a boy or a girl, it cannot be both, etc., etc. Three or more events originating from the same experiment are mutually exclusive if pair wise they are mutually exclusive. If the two events can occur at the same time, they are not mutually exclusive, e.g., if we draw a card from an ordinary deck of 52 playing cars, it can be both a king and a diamond. Therefore, kings and diamonds are not mutually exclusive. Similarly, inflation and recession are not mutually exclusive events. Speaking of playing cards, it is to be remembered that an ordinary deck of playing cards contains 52 cards arranged in 4 suits of 13 each. The four suits are called diamonds, hearts, clubs and spades; the first two are red and the last two are black. The face values called denominations, of the 13 cards in each suit are ace, 2, 3, …, 10, jack, queen and king. The face cards are king, queen and jack. These cards are used for various games such as whist, bridge, poker, etc. We have discussed the concepts of mutually exclusive events. Another important concept is that of exhaustive events.
EXHAUSTIVE EVENTS
Events are said to be collectively exhaustive, when the union of mutually exclusive events is equal to the entire sample space S.
EXAMPLES
• In the cointossing experiment, ‘head’ and ‘tail’ are collectively exhaustive events.
• In the dietossing experiment, ‘even number’ and ‘odd number’ are collectively exhaustive events. In conformity with what was discussed in the last lecture:
PARTITION OF THE SAMPLE SPACE
A group of mutually exclusive and exhaustive events belonging to a sample space is called a partition of the sample space. With reference to any sample space S, events A and ⎯A form a partition as they are mutually exclusive and their
A is shaded
Next, we consider the concept of equally likely events:
EQUALLY LIKELY EVENTS
Two events A and B are said to be equally likely, when one event is as likely to occur as the other. In other words, each event should occur in equal number in repeated trials.
EXAMPLE
When a fair coin is tossed, the head is as likely to appear as the tail, and the proportion of times each side is expected to appear is 1/2.
EXAMPLE
If a card is drawn out of a deck of wellshuffled cards, each card is equally likely to be drawn, and the probability that any card will be drawn is 1/52.
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