Bernoulli trials : When experiments are dichotomous in nature means there are only two outcomes then trials of a random experiment are called Bernoulli trials .

For example  : A tossed coin shows a head or tail , any manufactured item can be defective or non - defective and etc .

Conditions of Bernoulli trials

(a)   There should be a finite number of trials .
(b)    The probability of success remains the same in each trials .
(c)    The trials should should be independent .
(d)    Each trials has exactly two outcomes success or failure .

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Binomial Distribution  :
Binomial distribution is an discrete probability distribution . It is bi - parametric distribution , parameters are n (number of trials) and p (probability of success of random variable x).
It is denoted as x ∿ B (n,p).

If the values of parameters n and p are known then we can find the complete probability distribution . The probability of x successes in n trials P(X = x) is also denoted by P(x) and is given by

P(x) = ⁿC _x . p^x . q^{n - x} , x = 0 , 1, ---- n .

This P(x) is called the probability function of the binomial distribution .

→    Mean of the binomial distribution is = np

→   Variance of the binomial distribution distribution is = npq

→    Mean > variance or np > npq

→   The variance of x has maximum value n/4 at p = q = 0.5

→   Additive property of binomial distribution . If x and yare two independent variable such that
       x ∿ B(n₁ , p)
       y ∿ B(n₂ , P)
       Then (x + y) ∿ B(n₁ + n₂ , p).

→   If p > 1/2 it is negatively skewed distribution and when p < 1/2 , then it is positively skewed distribution .

→    If p = 1/2 it is symmetrical distribution .

→   When n is very large and p is very small then binomial distribution tends to possion distribution .

→    When n is very large , p and q are not small , then binomial distribution tends to normal distribution.

→    Recurrence relation for probability is given by

p(x + 1) =

n - x/x + 1

.

p/q

. p(x)

It is used to derive successive probabilities of binomial variate x.