Probability
Probability is a way of measuring uncertainity.
Example:
1. Tossing a coin
2. Result of a sporting event
3. Weather forecast.
Some important terms:
1. Trial: Each attempt of a random experiment is a trial.
Example:
In a random experiment of tossing a coin, if we have thrown the coin \(10\) times, then we have attempted \(10\) trials.
2. Outcome: The possible set of results at the end of each of the trials is an outcome.
Example:
Head is obtained as the outcome on the first attempt of tossing a coin.
3. Sample point: While a trial is conducted, each possible outcome is the sample point
Example:
The sample points while tossing a coin is Head and Tail.
The sample points while rolling a die is 1, 2, 3, 4, 5 and 6.
4. Sample space: The collection of sample points is a sample space. The sample points in sample space is enclosed in curly braces. Sample space is denoted by \(S\). The number of points in a sample space is denoted by \(n(S)\).
Example:
The samle space of tossing a coin:
\(S = {H, T}\)
\(n(S) = 2\)
5. Event: An event is a subset of a sample sapce.
Example:
The sample space of rolling a die, \(S ={1, 2, 3, 4, 5, 6}\)
An event of rolling a multiple of \(3\) is \({3, 6}\)
Introduction to classical approach:
The classical approach is the oldest appraoch of calculating probaility.
Equally likely:
When a set of objects are mixed, and one object has to be picked up at random, then all the objects have an equal probability of getting picked up on the first attempt.
For an event, \(E\) is the probability of getting a favourable outcome is given by:
The probability of event \(E\) is shown as \(P(E)\)
\(P(E) = \frac{\text{The number of favourable outcomes}}{\text{The number of elements in the sample space}}\)
\(P(E) = \frac{n(E)}{b(S)}\)
Important!
- The probability value would always be between \(0\) and \(1\).
- The probability of equally likely outcomes are always equal.
Empirical approach:
When a probability is found considering the previously observed data or based on a larger set of observations, it is called an empirical probability.
Example:
When a 20000 pairs of shoes are ordered from the company and 2500 of them are faulty, what is the number of faulty pair of shoes?
The empirical probability is considered in the case of a large number of trials. It is defined as:
\(P(E) = \frac{\text{Number of trials in which the event happened}}{\text{Total number of trials}}\)
Types of events:
1. Sure event or certain event: A certain event will have a probability of \(1\)
Example:
On rolling a die, the probability of getting a number lesser that \(7\) is a sure event.
2. Impossible event: An impossible event will have a probability of \(0\).
Example:
On rolling a die, the probability of getting a number greater than \(7\) is an impossible event.
3. Complementary event: For an event \(E\) let the outcome obtained for the event be \(A\). Also, let the outcome not accepted for the event be \(A'\). In this case, the complementary event to \(A\) is \(A'\).
If \(S\) is the sample space and \(A\) is the set of favourable outcomes of that event, then the complementary event \(A'\) is \(S-A\).
Example:
Consider the event of rolling a fair die.
The sample space, \(S = {1, 2, 3, 4, 5, 6}\)
The outcome of the event \(A\) is \({1, 2, 3}\)
Therefore, the outcome of its complementary event \(A' = S - A\)
\(={1, 2, 3, 4, 5, 6} - {1, 2, 3}\)
\( = {4, 5, 6}\).