The counting techniques in probability, statistics, mathematics, engineering, and computer science are essential tools. Counting Techniques in probability help in determining the number of ways a particular event can occur.
Table of Contents
The following are the most common counting techniques in probability theory:
Factorial
For any integer $n$, $n$ factorial (denoted by $n!$) is the descending product beginning with $n$ and ending with 1. It can be written as
$$n! = n\times (n-1) \times (n-2) \times \cdots \times 2 \times 1$$
The example of factorial counting are:
- $3! = 3\times 2\times 1 = 6$
- $5! = 5\times 4\times 3! = 20 \times 6 = 120$
- $10! = 10\times 9\times 8\times 7\times 6\times 5! = 3628800$
Note that a special definition is made for the case of $0!$, $0!=1$.
Permutations
A permutation of a group of objects is an ordered arrangement of the objects. The number of different permutations of a group of $n$ objects is $n!$. The formula of permutation is
$$P(n, r) = {}^nP_r = \binom{n}{r} = \frac{n!}{(n-r)!}$$
where $n$ is the total number of objects, and $r$ is the number of objects to be arranged.
The example of permutations are:
- The number of ways of dealing with the cards of a standard deck in some order is $52! = 8.066\times 10^{67}$
- Suppose, we want to place a set of five names in some order, there are five choices for which name to place first, then 4 choices of which to list second, 3 choices for third, 2 choices for fourth, and only one choice for the last (fifth one). Therefore, one can determine, how many different ways can 5 people be ordered in a row can be counted using the fundamental counting principle, the number of different ways to put 5 names in order is $5! = 5\times 4 \times 3\times 2\times 1 = 120$
Often entire set of objects is not required to be placed in order, usually one wants to compute how many ways a few chosen objects can be ordered. For example,
Example: A horse race has 14 horses, how many different possible ways can the top 3 horses finish?
Solution: There are 14 possibilities for which horse finishes first, 13 for second, and 12 for third. So, by the fundamental counting principle, there are $14\times 13\times 12 = 2184$ different possible ways (finishing orders) for the top three horses. $\binom{14}{3} = \frac{14!}{(14-3)!}=2184$.
In the above examples, permutations are called permutations of $n$ objects taken $r$ at a time.
Combinations
A combination is a selection of objects from a set without regard to order. Combinations are used when calculating the number of outcomes for experiments involving multiple choices, and often the order of the choices is not required. The formula of combination is
$$C(n, r) = {}^nC_r = \frac{n!}{r!(n-r)!}=\frac{{}^nP_r}{r!}$$
where $n$ is the total number of objects, and $r$ is the number of objects to be arranged without any regard to importance or order.
The example of combinations are:
- Drawing a 5-card poker hand (${}^{52}C_5$)
- Selecting a three-person committee from a group of 30 (without any priority or importance) (${}^{30}C_3$)
A choice of $r$ objects from a group of $n$ objects without regard to order is called a combination of $n$ objects taken $r$ at a time.
Example: In how many different ways can a committee of 3 people be chosen from a group of 10 people?
Solution: $C(10, 3) = \frac{10!}{3!(10-3)!} = 120 ways$
Multiplication Principle
If one event can occur $m$ times and another event occurs $n$ times, then the occurrence of the two events together can be computed using the multiplication principle, that is, by multiplying $m\times n$. For example, if there are 5 shirts and 3 pants to choose from, one can compute the different ways of outfits by multiplying the number of shirts and number of pants, i.e., $5 \times 3=15$, so there are 15 ways of outfits from 5 shirts and 3 pants.
Addition Principle
If one event can occur in $m$ ways and a second event can occur n $n$ ways, then one or the other event can occur in $m+n$ ways. For example, if there are 3 red balls and 4 blue balls, the number of was a ball can be chosen is: $4+3=7$.
Application of Counting Techniques in Probability
- Probability: Calculating probabilities of events based on the number of favorable outcomes and the total number of possible outcomes.
- Combinatorics: Studying the arrangement, combination, or selection of objects.
- Computer Science: Analyzing algorithms and data structures.
- Statistics: Sampling and hypothesis testing.
- Cryptography: Designing secure encryption methods
FAQs about Counting Techniques in Probability
- What is meant by counting techniques?
- What are the applications of counting techniques in probability?
- Define permutations and combinations.
- What is the difference between the multiplication and addition principles?
- Give real-life examples of permutations and combinations.
- Write down the formulas of permutations and combinations.
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