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

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

$\mu$ = 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:

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

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 Examples

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


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 $\times$ 0.00390625 $\times$ (1 – 0.5)(10 – 8) = 0.17578125 $\times$ 0.52 = 0.0439453125

The probability of getting 8 tails = 0.0439

Practise This Question

A bag contains 1 red ball 1 green ball 2 blue balls and 1 black ball. If a ball is drawn out of the bag, the probability of getting a ball of each colour is equally likely.