What does p represent in the binomial expansion equation?

What does p represent in the binomial expansion equation?

• A. Number of outcomes in one category
• B. Probability of success in one trial
• C. Total number of trials
• D. The combined probability of outcomes

Answer: (B)

In the context of binomial expansion, 'p' specifically represents the probability of success in a single trial. This is essential for calculations involving binomial distributions, which describe the number of successes in a fixed number of independent Bernoulli trials. The binomial formula is typically represented as \( (p + q)^n \), where \( p \) is the probability of success and \( q \) is the probability of failure (where \( q = 1 - p \)). In applying the binomial theorem, understanding the concept of success probability is crucial for determining how outcomes are distributed over repeated trials. This foundation allows us to calculate various probabilities, such as determining the likelihood of achieving a certain number of successes (k) in n trials, which is reflected in the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k q^{(n-k)} \] This formula depends heavily on accurately defining 'p' for the correct evaluation of outcomes under the specified conditions of the trials.

In the context of binomial expansion, 'p' specifically represents the probability of success in a single trial. This is essential for calculations involving binomial distributions, which describe the number of successes in a fixed number of independent Bernoulli trials. The binomial formula is typically represented as ( (p + q)^n ), where ( p ) is the probability of success and ( q ) is the probability of failure (where ( q = 1 - p )).

In applying the binomial theorem, understanding the concept of success probability is crucial for determining how outcomes are distributed over repeated trials. This foundation allows us to calculate various probabilities, such as determining the likelihood of achieving a certain number of successes (k) in n trials, which is reflected in the binomial probability formula:

[

P(X = k) = \binom{n}{k} p^k q^{(n-k)}

]

This formula depends heavily on accurately defining 'p' for the correct evaluation of outcomes under the specified conditions of the trials.