1. What is Binomial Distribution (Less Than)?

The binomial distribution calculates the probability of having fewer than k successes in n independent trials, each with success probability p. The "less than" version sums probabilities from 0 to k-1 successes.

2. How Does the Calculator Work?

The calculator uses the binomial probability formula:

Where:

• \( n \) — Number of trials
• \( k \) — Maximum number of successes (exclusive)
• \( p \) — Probability of success on each trial (0 ≤ p ≤ 1)
• \( \binom{n}{i} \) — Binomial coefficient "n choose i"

Explanation: The formula sums the probabilities of all outcomes with fewer than k successes.

3. When to Use Less Than Probability

Details: Use this when you need to know the probability of fewer than a certain number of successes. Common applications include quality control (defects below threshold), medical testing (fewer than x positive results), and risk assessment.

4. Using the Calculator

Tips: Enter positive integer for trials, non-negative integer for maximum successes (k), and probability between 0 and 1. The calculator will sum probabilities from 0 to k-1 successes.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between "less than" and "at most"?
A: "Less than k" means 0 to k-1 successes, while "at most k" means 0 to k successes (inclusive).

Q2: When is binomial distribution appropriate?
A: When trials are independent, have same success probability, and fixed number of trials.

Q3: What if k is 0?
A: P(X < 0) is always 0 since you can't have fewer than 0 successes.

Q4: How accurate is the calculation?
A: Very accurate for small n. For large n (>1000), normal approximation may be better.

Q5: Can I use this for continuous data?
A: No, binomial is for discrete counts. Use normal distribution for continuous data.