For a binomial experiment with n trials, p is the probability of success, q is the probability of failure, and x is the number of successes in n trials. Which one of the following statements is correct?
A) p + q = 1
B) p(x) = 1 for x = 0, 1, . . ., n
C) P(x = 0) =
D) p + q = 1 and p(x) = 1 for x = 0, 1, . . ., n
E) all of these

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The binomial distribution is a discrete probability distribution that describes the number of successes in a sequence of independent experiments, each of which has a constant probability of success. The probability of success is denoted by p, and the probability of failure is denoted by q. The total number of trials is denoted by n.The probability of obtaining x successes in n trials is given by the binomial probability mass function:```P(X = x) = (n! / x!(n - x)!) * p^x * q^(n - x)```where:* n is the number of trials* x is the number of successes* p is the probability of success* q is the probability of failureThe sum of the probabilities of all possible outcomes must equal 1. Therefore, we have:```∑[P(X = x)] = 1```This means that the sum of the probabilities of obtaining 0, 1, 2, ..., n successes must equal 1.The probability of obtaining 0 successes is given by:```P(X = 0) = (n! / 0!(n - 0)!) * p^0 * q^(n - 0) = q^n```Therefore, the probability of obtaining 0 successes is q^n.The probability of obtaining 1 success is given by:```P(X = 1) = (n! / 1!(n - 1)!) * p^1 * q^(n - 1) = n * p * q^(n - 1)```Therefore, the probability of obtaining 1 success is n * p * q^(n - 1).The probability of obtaining 2 successes is given by:```P(X = 2) = (n! / 2!(n - 2)!) * p^2 * q^(n - 2) = n * (n - 1) * p^2 * q^(n - 2)```Therefore, the probability of obtaining 2 successes is n * (n - 1) * p^2 * q^(n - 2).And so on.The probability of obtaining x successes in n trials is given by the binomial probability mass function. The sum of the probabilities of all possible outcomes must equal 1. The probability of obtaining 0 successes is q^n. The probability of obtaining 1 success is n * p * q^(n - 1). The probability of obtaining 2 successes is n * (n - 1) * p^2 * q^(n - 2). And so on.

For a Binomial Experiment with N Trials, P Is the Probability