Which of the following is the probability of x successes in a binomial distribution with number of trails n and probability of success as θ(0 < θ < 1) in each trail?

The correct option is:

$${}^n{c}_x{\theta }^x(1-\theta {)}^{n-x},x=0,1,2,...,n$$

Step-by-Step Explanation:

A binomial distribution describes the probability of obtaining a specific number of successes in a fixed number of independent trials (Bernoulli trials). Let us break down the formula components:

1. Definition of Variables:
- Let n be the total number of independent trials.
- Let θ be the probability of success in each individual trial, where 0 < θ < 1.
- The probability of failure in each individual trial is therefore 1 - θ.
- Let x be the number of successful trials, which can take any integer value from 0 up to n.

2. Probability of a Specific Sequence:
If we want exactly x successes and consequently n - x failures in a specific, ordered sequence, the probability of that sequence occurring is the product of the individual probabilities of each trial, since the trials are independent:
$${\theta }^x(1-\theta {)}^{n-x}$$

3. Accounting for All Possible Sequences:
There are multiple ways to arrange x successes and n - x failures in a sequence of n trials. The number of unique sequences is given by the combination formula:
$${}^n{c}_x=\frac{n!}{x!(n-x)!}$$

4. Final Probability Mass Function:
Multiplying the number of possible configurations by the probability of a single configuration yields the probability of obtaining exactly x successes:
$$P(X=x)={}^n{c}_x{\theta }^x(1-\theta {)}^{n-x}$$ where x = 0, 1, 2, ..., n. This matches the correct option.