Chapter 16

Probability

Probability and Random experiments:

A probability is simply a number between 0 and 1 that measures the uncertainty of a particular event.

Although many events are uncertain, we possess different degrees of belief about the truth of an uncertain event.

Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e. how likely they are to happen, using it. Probability can range in from 0 to 1, where 0 means the event to be an impossible one and 1 indicates a certain event.

OR

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty.

For example, most of us are pretty certain of the statement "the sun will rise tomorrow", and pretty sure that the statement "the moon is made of green cheese" is false.

For example, when we toss a coin, either we get Head OR Tail, only two possible outcomes are possible (H, T). But if we toss two coins in the air, there could be three possibilities of events to occur, such as both the coins show heads or both show tails or one shows heads and one tail, i.e.(H, H), (H, T),(T, T).

The Classical view of a Probability

Suppose that we observe some phenomena (say, the rolls of two dice) where the outcome is random. Suppose we can write down the list of all possible outcomes, and we believe that each outcome in the list has the same probability. Then the probability of each outcome will be

P(outcome)=1/(number of outcomes)

### The Frequency View of a Probability

The classical view of probability is helpful only when we can construct a list of outcomes of the experiment in such a way where the outcomes are equally likely.

The frequency interpretation of probability can be used in cases where outcomes are equally likely or not equally likely.

This view of probability is appropriate in the situation where we are able to repeat the random experiment many times under the same conditions.

An Experiment is defined as a process whose result is well defiened

Two types of Experiment:

1. Deterministic Experiment
2. Random Experiment

Deterministic Experiment:

It is an Experiment whose outcomes can be predicted with certainty under identical conditions.

Example: When we heat water it will evaporate

Example: When we toss a two headed coin we will get a head.

Random Experiment

It is an experiment whose all possible outcomes are known but it is not possible to predict the exact outcomes in advance.

Or

A random experiment is a mechanism that produces a definite outcome that  cannot be predicted with certainty.

Example: An unbiased coin is tossed.

Example: A die is rolled.

Random Experiment always satisfies the following two conditions:

1. It has more than one possible outcomes
2. It is not possible to predict the outcome in advance.

The two possible outcomes are getting a head or a tail. The outcome of this experiment cannot be predicted before it has been performed. Furthermore, it can be conducted many times under the same conditions. Thus, tossing a coin is an example of a random experiment.

Outcome: A possible result of a random experiment is called its outcome.