1. What is the Probability and Combination Formula?

The probability and combination formula calculates the probability of getting exactly b successes in a independent Bernoulli trials with success probability p. It combines combinatorial mathematics with probability theory.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( C(a,b) \) — Combination of a items taken b at a time
• \( p \) — Probability of success on a single trial (0 to 1)
• \( a \) — Total number of trials
• \( b \) — Number of successful trials

Explanation: The formula calculates the number of ways to get b successes (combination term) multiplied by the probability of each specific outcome with exactly b successes and (a-b) failures.

3. Importance of Probability Calculation

Details: This calculation is fundamental in statistics, quality control, risk assessment, and many scientific fields where predicting outcomes is essential.

4. Using the Calculator

Tips: Enter total trials (a), successful trials (b ≤ a), and probability (0 ≤ p ≤ 1). All values must be valid (positive integers for a and b, probability between 0 and 1).

5. Frequently Asked Questions (FAQ)

Q1: What is a Bernoulli trial?
A: A Bernoulli trial is a random experiment with exactly two possible outcomes: success (with probability p) or failure (with probability 1-p).

Q2: What's the difference between combination and permutation?
A: Combinations consider selection without regard to order, while permutations consider ordered arrangements.

Q3: What if b > a?
A: The probability is 0 since you can't have more successes than trials.

Q4: What does C(a,b) represent?
A: It's the binomial coefficient, calculated as a!/(b!(a-b)!), representing the number of ways to choose b items from a without regard to order.

Q5: When is this formula applicable?
A: When trials are independent with constant probability p, and when counting exact number of successes.