Probability Calculator With Numbers

Binomial Probability Formula:

\[ P = \binom{n}{r} \times p^r \times (1-p)^{n-r} \]

Number of Trials (n):

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Number of Successes (r):

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Probability of Success (p):

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Binomial Probability (P):

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1. What is Binomial Probability?

Binomial probability calculates the probability of getting exactly r successes in n independent Bernoulli trials, where each trial has the same probability of success p. It's used in statistics, genetics, quality control, and many other fields.

2. How Does the Calculator Work?

The calculator uses the binomial probability formula:

\[ P = \binom{n}{r} \times p^r \times (1-p)^{n-r} \]

Where:

\( n \) — Number of trials (must be positive integer)
\( r \) — Number of successes (must be between 0 and n)
\( p \) — Probability of success on a single trial (must be between 0 and 1)
\( \binom{n}{r} \) — Combination formula: \( \frac{n!}{r!(n-r)!} \)

Explanation: The formula calculates the probability of exactly r successes in n independent trials, where each trial has probability p of success.

3. Importance of Binomial Probability

Details: Binomial probability is fundamental in statistics for modeling binary outcomes. It's used in hypothesis testing, quality control, medical trials, and risk assessment where events have two possible outcomes.

4. Using the Calculator

Tips: Enter the number of trials (n), number of successes (r), and probability of success (p). All values must be valid (n ≥ 1, 0 ≤ r ≤ n, 0 ≤ p ≤ 1).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between binomial and normal distribution?
A: Binomial distribution is for discrete outcomes (success/failure), while normal distribution is for continuous data. For large n, binomial approximates normal distribution.

Q2: When is the binomial distribution appropriate?
A: When trials are independent, have only two outcomes, and the probability of success remains constant across all trials.

Q3: What if I need cumulative probability?
A: This calculator gives the probability of exactly r successes. For cumulative probability (r or fewer/more successes), you'd need to sum multiple binomial probabilities.

Q4: Are there limitations to binomial distribution?
A: It assumes independence between trials and constant probability of success, which may not hold in all real-world scenarios.

Q5: What is the expected value in binomial distribution?
A: The expected number of successes is n × p, and the variance is n × p × (1-p).