# Probability Distribution Formula

A table that assigns a probability to each of the possible outcome of a random experiment is a probability distribution equation. In simple words, it gives the probability for each value of the random variable.

There are two types of probability distribution:

**1. Normal Probability Distribution**

Its is also known as Gaussian distribution and it refers to the equation or graph which are bell shaped.

The formula for normal probability distribution is as stated

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

Where,

$\mu$ = Mean

$\sigma$ = Standard Distribution.

If mean($\mu$) = 0 and standard deviation($\sigma$) = 1, then this distribution is known to be normal distribution.

*x* = Normal random variable.

**2. Binomial Probability Distribution**

These are the probability occurred when the event consists of *n* repeated trials and the outcome of each trial may or may not occur.

The formula for binomial probability is as stated below:

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

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.

### Solved Examples

**Question 1: **Calculate the probability of getting 8 tails, if a coin is tossed 10 times ?

**Solution:**

Given,

Number of trails(n) = 10

Number of success(r) = 8(getting 8 tails)

probability of single trail(p) = $\frac{1}{2}$ = 0.5

To find _{n}C_{r} = $\frac{n!}{r!(n – r)!}$ = $\frac{10!}{8!(10 – 8)!}$ = $\frac{10 \times 9 \times 8!}{8!2!}$ = 45

To find p^{r} = 0.5^{8} = 0.00390625

So, probability of getting 8 tails

P(x) = _{n}C_{r }p^{r} (1- p)^{n – r} = 45 $\times$ 0.00390625 $\times$ (1 – 0.5)^{(10 – 8)} = 0.17578125 $\times$ 0.5^{2} = 0.0439453125

The probability of getting 8 tails = 0.0439

More topics in Probability Distribution Formula | |

Binomial Distribution Formula | Normal Distribution Formula |

Poisson Distribution Formula |

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