In binomial distribution formula why we use (1-p)^n-x.
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The binomial distribution formula is used to calculate the probability of obtaining a certain number of successes in a fixed number of independent trials, given a specified probability of success in each trial. The formula is:
P(X = x) = (n choose x) * p^x * (1-p)^(n-x)
where:
The term (1-p)^(n-x) is included in the formula to represent the probability of obtaining (n-x) failures in the remaining (n-x) trials after x successes have been obtained. This is necessary because the probability of obtaining x successes in n trials is equal to the probability of obtaining (n-x) failures in the remaining trials.
Therefore, the term (1-p)^(n-x) represents the probability of obtaining the remaining failures in the remaining trials, given that x successes have already occurred. This is multiplied by p^x, which represents the probability of obtaining x successes in the first x trials. Finally, the binomial coefficient (n choose x) is multiplied to account for the different ways in which x successes can occur in n trials.
here we are finding we are finding the probability of x successes in n trials. so p^x part is used to calculate the successful outcomes results and (1-p)^n-x part is used to calculate the unsuccessful outcomes result