1. What is Binomial Probability Between Two Numbers?

The binomial probability between two numbers calculates the probability of getting between a and b successes (inclusive) in n independent trials, each with success probability p. It's fundamental in statistics for binary outcome scenarios.

2. How Does the Calculator Work?

The calculator uses the binomial probability formula:

Where:

• \( n \) — Number of trials
• \( p \) — Probability of success on each trial
• \( a \) — Lower bound of successes
• \( b \) — Upper bound of successes
• \( C(n,k) \) — Combinations of n items taken k at a time

Explanation: The formula sums the probabilities of all possible outcomes between a and b successes.

3. Importance of Binomial Calculation

Details: This calculation is essential for hypothesis testing, quality control, and any scenario involving repeated independent trials with binary outcomes.

4. Using the Calculator

Tips: Enter positive integer for n (1-1000), probability p (0-1), and integers a and b between 0 and n with a ≤ b. Results are accurate to 6 decimal places.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between this and exact binomial probability?
A: This calculates cumulative probability between two values, while exact probability is for a specific count.

Q2: When should I use normal approximation instead?
A: When n is large (typically >30) and p is not too close to 0 or 1, normal approximation may be simpler.

Q3: What if a = b?
A: The calculation reduces to the exact binomial probability for that specific value.

Q4: How accurate are the results?
A: Very accurate for n ≤ 1000. For larger n, numerical precision may become an issue.

Q5: Can I calculate one-tailed probabilities?
A: Yes, set a=0 for lower tail or b=n for upper tail probabilities.