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Probability

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Probability theory began in the seventeenth century by the two great French mathematicians, Blaise Pascal and Pierre de Fermat over two problems from games of chance. They solved problems and influenced early researchers in establishing a mathematical theory of probability.

Probability refers to the prediction of occurring of an event and is easy to understand and calculate. It is a measure of how likely it is that something will happen or that a statement is true. Applications of probability can be seen in everyday life, risk assessment, reliability etc. Due to its immense success and wide applications, theory of probability is viewed as the most important area of mathematics.

Definition of Probability

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Probability is a branch of mathematics that deals with the likelihood that a particular event will happen in the future. It is expressed as a number between 1 and 0 and can be expressed in fractions, ratios, percentage.

An event with probability of 1 is said to be certain. An event with a probability of 0.5 is considered to have equal odds of occurring or not occurring and an event with probability of 0 is said to be an impossible event.

Probability Terms

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Given below are the important terms considered in probability.

Almost Sure: An event happens almost surely, if it happens with probability one.
Conditional Probability: Conditional probability is the probability that an event will occur, when another event is known to occur or to have occurred.
Conditional Expectation: Conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution.
Event: An event is a set of outcomes to which a probability is assigned.
Experiment: Any procedure which is infinitely repeated and has a well defined set of outcomes is known as the sample space.
Law of Large Numbers: A rule that assumes that as the number of samples increases, the average of these samples is likely to reach the mean of the whole population.
Mutually Exclusive: Two events are mutually exclusive, if they cannot occur at the same time.
Random Variable: Random variable is a variable whose value is subject to variations due to chance.
Sample Space: Sample space is the set of all possible outcomes.
Outcome: Something that follows from an action.
Variance: Measure of how far a set of numbers is spread out.

Probability Formulas

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Given below are the important formulas of probability.

Probability = P(A) = $\frac{\text{Number of outcomes favorable to event A}}{\text{Total number of outcomes}}$

Addition Rule:
P(A $\cup$ B) = P(A) + P(B) - P(A $\cap$ B), P(A$\cap$ B) $\neq$ 0

Bayes Formula:
P($\frac{A}{B}$) = P($\frac{B}{A}$) $\times$ ($\frac{P(A)}{P(B)}$)

Complementary Events:
P(A$^{c}$) + P(A) = 1

Cumulative Distribution Function:
$F_{X}(x)$ = P(X $\leq$ x)

Independent Events:
P(A $\cap$ B) = P(A). P(B)

Dependent Events:
P(A $\cap$ B) = P(A) $\times$ P($\frac{B}{A}$)

Multiplication Rule:
For independent events, P(A $\cup$ B) = P(A) $\times$ P(B)

Rules of Probability

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Some of the rules of probability are as follows:
  1. Any probabilities assigned must be a non negative number.
  2. Impossible event has a probability of 0.
  3. For any event E, 0 $\leq$ P(E) $\leq$ 1.
  4. Probability of sample space should be equal to 1.
  5. When two outcomes cannot happen at the same time, probability that either outcome occurs is the sum of the probabilities of the individual outcomes.

Types of Probability

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There are four types of probability as follows:
  1. Classical Probability
  2. Experimental Probability
  3. Theoretical Probability
  4. Subjective Probability

Classical Probability: Events and outcomes in sample space are determined by the rules of game. Given n equally likely outcomes, let 's' represent the number of successful outcomes and 'f' represent the number of failure outcomes, then s + f = n. The probability of success is $\frac{s}{n}$ and the probability of failure is $\frac{f}{n}$. Probability of success plus the probability of failure is equal to 100% or 1. Therefore, $\frac{s}{n}$ + $\frac{f}{n}$ = 1.

Experimental probability:
This is based on the number of possible outcomes by the total number of trials. Example: For a coin tossing experiment, when a coin is tossed, the total possible outcomes are head and tail. Total number of trials will be determined by the total number of times the coin is flipped. If a coin is flipped 25 times and it lands on head 17 times, then the probability is $\frac{17}{25}$.

Theoretical Probability: The theoretical probability P(E) of an event E is the fraction of number of times we expect E to occur, if we repeat the same experiment over and over.
Example: Roll a fair die. The probability of rolling 1 is one sixth of the time.
=> P(1) = $\frac{1}{6}$
Similarly, P(2) = $\frac{1}{6}$, ......., P(6) = $\frac{1}{6}$

Subjective Probability: Subjective probability is an individual person's measure of belief that an event will occur. Subjective probabilities contain no formal calculations and only reflect the subject's opinions and past experience and they differ from person to person. There will be a high degree of personal bias.

Examples of Probability

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Given below are some of the examples in probability.

Solved Examples

Question 1: A bag contains 3 red, 5 black, 7 white and 6 orange balls. If a ball is chosen at random from the bag, what is the probability that it is red or black?
Solution:
Number of balls = 3 + 5 + 7 + 6 = 21

P(red) = $\frac{3}{21}$$\frac{1}{7}$

P(black) = $\frac{5}{21}$

P(red or black) = P(red) + P(black)

 = $\frac{1}{7}$  +  $\frac{5}{21}$

= $\frac{8}{21}$

Therefore, the probability that the ball drawn at random from the bag is $\frac{8}{21}$.

Question 2: When two standard dice are rolled, what is the probability that the sum of the numbers on the top faces is 5.
Solution:
When two standard dice are rolled the outcomes are
A = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6)
     (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)
     (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)
     (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)
     (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)
     (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
As we see, there are 36 possible outcomes, when two standard dice are rolled. The outcomes in which the sum of the top faces is 5 are,
{(1, 4), (2, 3), (3, 2), (4, 1)}

P(sum of 5) = $\frac{\text{Number of successful outcomes}}{\text{Number of total outcomes}}$ = $\frac{4}{36}$
= $\frac{1}{9}$