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 table. In simple words, it gives the probability for each value of the random variable.

Formulas for Probability Distribution

The formulas for two types of the 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:

\(\large P(x)=\frac{1}{\sqrt{2\pi \sigma ^{2}}} e^{-(x-\mu )^{2}/2\sigma ^{2}}\)

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

It is defined as the probability that 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:

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

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 Questions

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, the probability of getting 8 tails is:

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

Question 2: Find the probability of normal distribution with population mean 2, standard deviation 3 of random variable 5.

Solution:
Given,
x = 5
Mean = μ = 2
Standard deviation = σ = 3
Normal probability distribution:
\(P(x)=\frac{1}{\sqrt{2\pi \sigma^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}\) \(P(x)=\frac{1}{\sqrt{2\times 3.14 \times 3^2}}e^{\frac{-(5-2)^2}{2\times 3^2}}\\=\frac{1}{\sqrt{56.52}}e^{\frac{-9}{2\times 9}}\\=\frac{1}{7.518}e^{\frac{-1}{2}}\\=\frac{0.6065}{7.518}\\=0.0807\)

Question 3: The probability of a man hits the target is ¼. If he fires 9 times, then find the probability that he hits the target exactly 4 times.

Solution:
Number of fires = n = 9
Number of success hits = r = 4
Probability of hitting the target = p = ¼
Probability of not hitting the target = q = 1 – p = 1 – (¼) = ¾
Finding nCr :
9C4 = 9!/[4! (9-4)!] = 9!/(4! 5!) = (9 × 8 × 7 × 6 × 5!)/(4 × 3 × 2 × 1 × 5!) = 126
Probability of the person hits the target exactly 4 times
= 9C4 (¼)4(¾)(9-4)
= 126 × (1/256) × (243/1024)
= 0.1168

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