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.
- _{n}C_{r} = $\frac{n!}{r!(n – r)!}$
- 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 _{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 × 0.00390625 × (1 – 0.5)^{(10 – 8)} = 0.17578125 × 0.5^{2} = 0.0439453125
The probability of getting 8 tails = 0.0439
Topics Related to Probability Distribution Formula | |
---|---|
Binomial Distribution Formula | Normal Distribution Formula |
Poisson Distribution Formula | Gaussian Distribution Formula |