Conditional Probability is the probability of one event occurrence having the same relationship with other events too. For example –

- There is an Event A and it states that it is raining outside. The probability of the event is 0.3% or 30 percent.
- There is another event B that states you will not go outside if it is raining. The probability of the event is 0.5% i.e. 50 percent.

Look at these events carefully having a direct relationship to each other. If it will be raining then you are not supposed to go outside in the case. The event B will happen when A has occurred already. If A will not happen then B will also not occur automatically. This probability is written P(B|A), the notation for the probability of B given A.

When A and B events are not related to each other somehow then they are simply written as P(B), P(A).From the above definition, the conditional probability could be calculated quickly and it is given by the formula –

When B is given by A, then conditional probability is,

\[\ P(B|A)= \frac{P(A\cap B)}{P(A)}\]

When A is given by B, then the conditional probability is

\[\ P(A|B)= \frac{P(A\cap B)}{P(B)}\]

The formula can be applied successfully only if the value of P(A) is greater than zero. The conditional probability could have many applications in the real-life like finance sector, insurance, or politics etc. The election process is an example of conditional probability where one can win only if the opponent fails.

The binomial probability is simply thought of as the probability of success or failure outcomes during an experiment or survey which are related somehow. This is also named as the binomial distribution with chances of two possible outcomes. Take an example of the coin tossed in the air has only two outcomes i.e. Head or Tail. At the same time, when a student appears for an exam, there are two possibilities again i.e. Pass or Fail. The binomial distribution should satisfy the three criteria, as given below –

- The number of trials or observations will be fixed and you could calculate the probability of something happening multiple numbers of times. So, chances for one outcome is 50 percent every time in this case.
- The number of trails or observations should be independent without impacting each other.
- The probability of success is equal to the probability of failure. Once you know the binomial distribution and its formula then you can apply the same to solve the complex problems in mathematics.

\[\ P(X) = C_{x}^{n} P^{x} q^{n-x}\]

Where,

n = Total number of trials

x = Total number of successful trials

p = probability of success in a single trial

q = probability of failure in a single trial = 1-p

A probability distribution tells you about the probability of the event occurrence and it can be used for such complex systems like the success rate of a drug during the cancer treatment. It can be represented with the help of a table in mathematics. The sum of probability distributions should always be 100 percent or 1 in the decimal.

\[\large p(x)=\frac{1}{\sqrt{2\pi \sigma^{2}}}\;e^{\frac{(x-\mu)^{2}}{2\sigma^{2}}}\]

Where,

μ = Mean

σ = Standard Distribution.

If mean(μ) = 0 and standard deviation(σ) = 1, then this distribution is known to be normal distribution.

*x* = Normal random variable.

\[\large p(x)=\frac{n!}{r!(n-r)!}\cdot p^{r}(1-p)^{n-1}=C(n,r)\cdot p^{r}(1-p)^{n-r}\]

Where,

n = Total number of events

r = Total number of successful events.

p = Probability of success on a single trial.

_{n}C_{r} = \[\ \frac{n!}{r!(n – r)!}\]

1 – p = Probability of failure.

The other name for empirical probability is experimental probability to calculate the probability of an experiment and a certain result too. It is usually required during the survey when the experiment is conducted over 100 people or more and give educational data accordingly.

\[\ P(E)=\frac{Number\;of\;times\;event\;occurs}{Total\;number\;of\;times\;experiment\;performed}\]

\[\ P(E)=\frac{f}{n}\]

**P(E)** = Empirical Probability