# Probability it as a tool which allows us

Probability is a statistical tool we should
understand how it works to use it correctly.

Probability is related to statistics in an
important way .we can use it as a tool which allows us to evaluate the
reliability of our conclusions about the population.

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Definition of probability :

Probability is the chance that some events
will occur sometimes it can be expressed by numbers or by words such as
impossible and unlikely.

Probability theorems:

There are ninety four theorems of
probability but we are going to discuss the most common and important  two theorems in the next few paragraphs.

1- Chebyshev’s
inequality :

This theorem is also known as tchebysheff’s   inequality. It is usually stated for random
variables but it can also be stated for measure spaces.   Chebyshev’s Theorem is used to find the proportion of observations you would
expect to find within two standard deviations from the mean.

Example:  a left-skewed distribution has a mean of 4.99 and a standard deviation of 3.13

To calculate it you should:

Step 1: Square the number of standard deviations:
22 = 4.

1 / 4 = 0.25.

Step 3: Subtract Step 2 from 1:
1 – 0.25 = 0.75.

At least 75% of
the observations fall between -2 and +2 standard deviations from the mean.
That’s:
mean – 2 standard deviations
4.99 – 3.13(2) = -1.27
mean + 2 standard deviations
4.99 + 3.13(2) = 11.25
Or between -1.27 and 11.25

2- bayes’ theorem:

It is a description to update hypotheses
probabilities if there is en evidence

It
depends on the axioms of conditional probability. It is used to find the
reverse probability.

Example:

you want to test a patient’s probability of
having lung cancer if they are a smoker so
(being a smoker) is the test for lung cancer.

·
A  could
mean (the patient has lung cancer ). Past data tells that 10% of the patients
have lung cancer P(A) = 0.10

·
B could mean ( the patient is a smoker ). 5% of
the patients are smokers  P(B) = 0.05.

You also know that among the
patients diagnosed with lung cancer  7%
are smokers

This is your B|A: the
probability that a patient is a smoker, given that they have lung cancer  is 7%.

Bayes’ theorem tells that:
P(A|B) = (0.07 * 0.1)/0.05 = 0.14
In other words, if the patient is a smoker, their chances of having lung cancer is 0.14 (14%) . But it’s
still unlikely that any particular patient has lung cancer.

Types of random variables:

A variable
X is a random variable if the value that assumes corresponding to the outcome
of an experiment is a chance or random event .

It is known that there are two types of random variables:

1-
Discrete random variables

2-
Continuous random variables

First
let’s have a look on the first type of random variables which is discrete
random variables

In a class experiment, a
student may be chosen randomly and one of the random variables can be the chemistry
grade. Probability distribution is associated with random variables it can limits the value in any subset of possible values such
as (more than 90 and less than 95) or (equals 50 or more than 50)

Example: coin throw – dice roll.

The second type of random variables is continuous random
variables

It is a type of random variables where the data can be more than
one value for example: in a running competition if you want to measure the time
taken to run 3 km it is continuous as there is more than one time can be taken.