What Is Binomial With Steps?
Most binomial calculators take n, p and k and return a probability. This one runs the other way: give it a target and it finds the k, the mode or the n that meets it.
Choose a percentile search to find the smallest k whose cumulative probability reaches your target. Choose the mode to find the most likely number of successes. Choose find n to get the fewest trials that give at least k successes with a chosen probability.
Every mode shows its steps, a highlighted chart and, where useful, a probability table around the centre.
How to Use the Binomial With Steps Calculator
- Choose what to find.
- Enter n, p and the target as needed.
- Read the answer and each step.
- Check the highlighted chart and the table.
Binomial With Steps Formula
Binomial probability
Mode
Binomial With Steps Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| n | Number of trials | |
| p | Probability of success | |
| k | Number of successes | |
| P(X ≤ k) | Cumulative probability |
Binomial With Steps Worked Example
With n = 40 and p = 0.3, the mean is 12. The mode is ⌊41 × 0.3⌋ = 12. The 90th percentile is the smallest k where P(X ≤ k) reaches 0.9, which is 16. To get at least one success with probability 0.95 when p = 0.3, solve 1 − 0.7ⁿ ≥ 0.95, which gives n = 9.
With the example values loaded in the calculator above, the steps are:
- We add P(X = 0), P(X = 1), … until the running total first reaches 0.9.
- P(X ≤ 15) = 0.884853 is still below 0.9.
- P(X ≤ 16) = 0.936687 reaches it, so k = 16 is the 90th percentile.
Result: Smallest k with P(X ≤ k) ≥ 0.9 = 16.
How to Read These Inverse Results
These modes work backwards from a probability. With n = 40 and p = 0.3 the mean is 12 and the mode is ⌊41 × 0.3⌋ = 12. The 90th percentile is the smallest k with P(X ≤ k) ≥ 0.9, which is 16, because P(X ≤ 15) = 0.885 is still below 0.9.
Because the binomial is discrete, the cumulative probability usually overshoots the target a little, here 0.937 at k = 16. For the "how many trials" mode, the page gives the smallest n that meets the target and the probability at n − 1, which falls just short.
Where Binomial With Steps Is Used
Percentiles
Find the k that 90% or 95% of outcomes fall below.
Most likely outcome
Find the mode of a binomial.
Sample size for a target
Find the fewest trials to reach a chosen chance of at least k successes.
Checking steps
Follow the cumulative sums or the search for n.
Common Binomial With Steps Mistakes to Avoid
- The percentile is a whole number.
- If (n + 1)p is a whole number, two values tie for the mode.
- "At least one success" needs n ≥ ln(1 − target) ÷ ln(1 − p).
- Trials must be independent with constant p.
Binomial With Steps Terms and Definitions
- Percentile (quantile)
- The smallest k at which the cumulative probability reaches the target.
Binomial With Steps FAQs
How do you find the most likely binomial outcome?
Take the floor of (n + 1)p. If (n + 1)p is a whole number, both that value and the one below it are equally likely.
How many trials do I need for at least one success?
Solve 1 − (1 − p)ⁿ ≥ target, which gives n ≥ ln(1 − target) ÷ ln(1 − p).
What is the inverse binomial?
It is the k at which the cumulative probability first reaches a chosen level.