probability total details of probability formula

1. Probability

2. meaning
Probability means possibility. It is a branch of mathematics that deals with
the occurrence of a random event. The value is expressed from zero to one.
Probability has been introduced in Maths to predict how likely events are to
happen.

3. DEFINATION
 Probability is a mathematical term people use for the likelihood that an event
will happen, like rolling a two with a die or drawing a king from a deck of cards.
Whether you're aware of it, you use probability every day when making
decisions about events with an uncertain outcome, from playing games to
choosing an insurance policy.
 Probability And Statistics are the two important concepts in Maths. Probability is
all about chance. Whereas statistics is more about how we handle various data
using different techniques. It helps to represent complicated data in a very
easy and understandable way.

5. PROBABILITYDISTRIBUTION
In Statistics, the probability distribution gives the possibility of each outcome of
a random experiment or event. It provides the probabilities of different possible
occurrences.
To recall, the probability is a measure of uncertainty of various phenomena.
Like, if you throw a dice, the possible outcomes of it, is defined by the
probability. This distribution could be defined with any random experiments,
whose outcome is not sure or could not be predicted. Let us discuss now its
definition, function, formula and its types here, along with how to create a
table of probability based on random variables.

6. Types of distribution
 Bernoulli distribution
 Normal distribution,
 chi-square distribution,
 binomial distribution,
 uniform distribution
Are some of the many different classifications of probability distributions.

7. Bernoulli distribution
 A Bernoulli distribution has only two bernoulli trials or possible outcomes,
namely 1 (success) and 0 (failure), and a single trial. So the random variable X
with a Bernoulli distribution can take the value 1 with the probability of
success, say p, and the value 0 with the probability of failure, say q or 1-p.
 Here, the occurrence of a head denotes success, and the occurrence of a tail
denotes failure.
Probability of getting a head = 0.5 = Probability of getting a tail since there
are only two possible outcomes.
 The probability mass function is given by: px(1-p)1-x where x € (0, 1)
 It can also be written as:

8. NORMAL DISTRIBUTION
 The normal distribution represents the behavior of most of the situations in the universe
(That is why it’s called a “normal” distribution. I guess!). The large sum of (small) random
variables often turns out to be normally distributed, contributing to its widespread
application. Any distribution is known as Normal distribution if it has the following
characteristics:
 The mean, median, and mode of the distribution coincide.
 The curve of the distribution is bell-shaped and symmetrical about the line x=μ.
 The total area under the curve is 1.
 Exactly half of the values are to the left of the center, and the other half to the right.

9. chi-square
 A chi-square (Χ2) distribution is a continuous probability distribution that is
used in many hypothesis tests.
 The shape of a chi-square distribution is determined by the parameter k. The
graph below shows examples of chi-square distributions with different values
of k.
Formula Explanation
Where
•X² is the chi-square test statistic
• is the summation operator (it means
“take the sum of”)
• is the observed frequency
• is the expected frequency

10. Binomial Distribution
 A distribution where only two outcomes are possible, such as success or failure,
gain or loss, win or lose and where the probability of success and failure is the
same for all the trials is called a Binomial Distribution.
 Based on the the properties of a Binomial Distribution are:
 Each trial is independent.
 There are only two possible outcomes in a trial – success or failure.
 A total number of n identical trials are conducted.
 The probability of success and failure is the same for all trials. (Trials are
identical.)
 The mathematical representation of binomial distribution is given by:


11. Uniform Distribution
 When you roll a fair die, the outcomes are 1 to 6. The probabilities of getting
these outcomes are equally likely, which is the basis of a uniform distribution.
Unlike Bernoulli Distribution, all the n number of possible outcomes of a
uniform distribution are equally likely.
 A variable X is said to be uniformly distributed if the density function is:
 F(x)=1/b-a

12. ADDITION THEROEM
 The probability of happening an event can easily be found using the definition
of probability. But just the definition cannot be used to find the probability of
happening at least one of the given events. A theorem known as “Addition
theorem” solves these types of problems. The statement and proof of
“Addition theorem” and its usage in various cases is as follows.
 Mutually exclusive events:
 Two or more events are said to be mutually exclusive if they don’t have any
element in common. i.e. if, the occurrence of one of the events prevents the
occurrence of the others then those events are said to be mutually exclusive.
 P(A or B)= P(A)+P(B)

13. MULTIPLICATION THEOREM

14. In conditional probability, we know that the probability of
occurrence of some event is affected when some of the
possible events have already occurred. When we know that a
particular event B has occurred, then instead of S, we
concentrate on B for calculating the probability of
occurrence of event A given B.
Taking the above example of throwing of two dice, the
possible outcomes are
S = {(x, y): x, y = 1, 2, 3, 4, 5, 6}.

15. Basics concepts of probability

16. Probability is a numerical measure of the likelihood that an
event will occur. The probability of an event is the long-
term relative frequency of that event. Probabilities are
numbers between zero and one, inclusive—that is, zero and
one and all numbers between these values.
Classical Method Approach to Probability
Empirical Method Approach to Probability
Subjective Method Approach to Probability