Probability Theory

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I. Probability Theory
* A branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance. * The word probability has several meanings in ordinary conversation. Two of these are particularly important for the development and applications of the mathematical theory of probability. One is the interpretation of probabilities as relative frequencies, for which simple games involving coins, cards, dice, and roulette wheels provide examples. * It is the likeliness of an event happening based on all the possible outcomes. The ratio for the probability of an event 'P' occurring is P (event) = number of favorable outcomes divided by number of possible outcomes. Example:

A coin is tossed on a standard 8×8 chessboard.
What is the theoretical probability that the coin lands on a black square?

Choices:
A. 0.5
B. 0.25
C. 0.42
D. 0.6

Correct answer: A

Solution:

Step 1: Theoretical probability = number of favorable outcomes / number of possible outcomes.

Step 2: The probability of the coin lands on the black square is 32.

Step 3: Total number of outcomes = 64.
Step 4: P (event) = 

Step 5: == 0.5

Step 6: The theoretical probability that the coin lands on a black square is 0.5.      

A. Permutation and Combination

* The various ways in which objects from a set may be selected, generally without replacement, to form subsets. This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor. By considering the ratio of the number of desired subsets to the number of all possible subsets for many games of chance in the 17th century * The French mathematicians Blaise Pascal and Pierre de Fermat gave impetus to the development of combinatorics and probability theory. * Permutation: Permutation means arrangement of things. The word arrangement is used, if the order of things is considered. * Combination: Combination means selection of things. The word selection is used, when the order of things has no importance.

Example of Permutation:
1. Suppose we have to form a number of consisting of three digits using the digits 1,2,3,4, to form this number the digits have to be arranged. Different numbers will get formed depending upon the order in which we arrange the digits.  2. How many different signals can be made by 5 flags from 8-flags of different colors?

Ans.    Number of ways taking 5 flags out of 8-flage = 8P5

=   8! / (8-5)!       

= 8 x 7 x 6 x 5 x 4 = 6720

3. Q. How many words can be made by using the letters of the word “SIMPLETON” taken all at a time?

Ans.   There are ‘9’ different letters of the word “SIMPLETON”

Number of Permutations taking all the letters at a time = 9P9

= 9!    = 362880.

Example of combination:
1. Now suppose that we have to make a team of 11 players out of 20 players, because the order of players in the team will not result in a change in the team. No matter in which order we list out the players the team will remain the same! For a different team to be formed at least one player will have to be changed.

2. Find the number of different choices that can be made from 3 apples, 4 bananas and 5 mangoes, if at least one fruit is to be chosen.

Ans:

Number of ways of selecting apples = (3+1) = 4 ways.
Number of ways of selecting bananas = (4+1) = 5 ways.
Number of ways of selecting mangoes = (5+1) = 6 ways.
Total number of ways of selecting fruits = 4 x 5 x 6
But this includes, when no fruits i.e. zero fruits is selected => Number of ways of selecting at least one fruit = (4x5x6) -1 = 119

B. Types of Probability
Classical theory of probability
* The classical approach to probability is to count the number of favorable outcomes, the number of total...
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