Probability Distribution Formula

Probability distribution formula mainly refers to two types of probability distribution which are normal probability distribution (or Gaussian distribution) and binomial probability distribution. To recall, a table that assigns a probability to each of the possible outcomes of a random experiment is a probability distribution equation. In simple words, it gives the probability for each value of the random variable.

Formulas for Probability Distribution

The formulas for two types of probability distribution are:

Normal Probability Distribution Formula

It 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,

  • μ = Mean
  • σ = Standard Distribution.
  • x = Normal random variable.

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

Binomial Probability Distribution Formula

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:

\(\large \large p(x)=\frac{n!}{r!(n-r)!}\cdot p^{r}(1-p)^{n-1}= _{n}C_{r}\cdot p^{r}(1-p)^{n-r}\)

Where,

  • n = Total number of events
  • r = Total number of successful events
  • p = Probability of success on a single trial.
  • nCr = $\frac{n!}{r!(n – r)!}$
  • 1 – p = Probability of failure.

Solved Probability Distribution Example Question

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 nCr = $\frac{n!}{r!(n – r)!}$ = $\frac{10!}{8!(10 – 8)!}$ = $\frac{10 \times 9 \times 8!}{8!2!}$ = 45

To find pr = 0.58 = 0.00390625

So, probability of getting 8 tails

P(x) = nCr pr (1- p)n – r = 45 × 0.00390625 × (1 – 0.5)(10 – 8) = 0.17578125 × 0.52 = 0.0439453125

The probability of getting 8 tails = 0.0439


Practise This Question

A solid cylinder of mass 2 kg and radius 0.2 m is rotating about its own axis without friction with angular velocity 3 rad/s. A particle of mass 0.5 kg and moving with a velocity 5 m/s strikes the cylinder and sticks to it as shown in figure below. The angular momentum of the cylinder before collision will be