Probability Calculator: Binomial, Normal & Poisson Distribution Tool

How to use the tool

1. Select a distribution. Pick Binomial, Normal or Poisson from the menu.
2. Fill every field. The calculator checks ranges automatically—no negative σ or λ.
3. Hit “Calculate”. Your probability appears below the form together with an interactive chart.
4. Download or screenshot the chart for reports or presentations.

Binomial fields

• Number of trials (n) – whole number ≥ 1. Examples: 15 coin flips, 60 manufactured parts.
• Probability of success (p) – 0 ≤ p ≤ 1. Examples: 0.42 defect rate, 0.95 pass probability.
• Number of successes (k) – 0 ≤ k ≤ n. Examples: exactly 6 heads, 54 good parts.

Normal fields

• Mean (μ) – any real value. Examples: 72 kg weight, 12 V voltage.
• Standard deviation (σ) – positive. Examples: 6 kg, 0.3 V.
• Value (X) – point for cumulative probability. Examples: 80 kg, 11.7 V.

Poisson fields

• Average rate (λ) – positive. Examples: 3.8 emails/min, 12 hits/hour.
• Occurrences (k) – whole number ≥ 0. Examples: 2 emails, 15 hits.

Formulas

• Binomial: $$P(X=k)=rac{n!}{k!(n-k)!}\,p^{k}(1-p)^{n-k}$$
• Normal (cumulative): $$P(X\le x)=\Phi\!\left(rac{x-\mu}{\sigma}\right)$$ where Φ is the standard-normal CDF.
• Poisson: $$P(X=k)=rac{\lambda^{k}e^{-\lambda}}{k!}$$

Example calculations

• Binomial: n = 12, p = 0.3, k = 4 → P = 0.231 (Stat Trek, 2023).
• Normal: μ = 50, σ = 5, X = 60 → P(X ≤ 60) = 0.977 2 (NIST Z-table, 2013).
• Poisson: λ = 4.5, k = 2 → P ≈ 0.113 (Walpole & Myers, 2012).

Quick-Facts

• Binomial variance equals n p (1 − p) (NIST Handbook 151, 2014).
• Φ(1.96) = 0.975; used for 95 % confidence limits (ISO 3534-1, 2006).
• Poisson mean equals its variance λ (Stat Trek, 2023).
• σ must be > 0; zero leads to undefined z-scores (Montgomery, 2020).

FAQ

What is the difference between cumulative and exact probability?

Cumulative sums probabilities up to a point (≤ k or ≤ X); exact gives only P(X = k). Exact suits “exactly five defects,” cumulative suits “at most five defects” (Moore & McCabe, 2021).

Can I input decimal successes in the Binomial field?

No. k must be an integer because successes are countable events (NIST Handbook 151, 2014).

Why does the Normal option return a value near 0.5 when X equals μ?

The CDF at the mean of a symmetric normal distribution is 0.5 by definition (ISO 3534-1, 2006).

How do I interpret a Poisson probability of 0.113?

You expect that outcome roughly 11 % of intervals; “events are rare yet possible,” notes Walpole & Myers (2012).

What range of λ is practical?

Models work well for λ ≤ 30; above this Poisson approximates Normal with μ = λ and σ ≈ √λ (Ross, 2022).

Do factorials limit n in the Binomial formula?

Large n cause overflow in direct factorials; the tool uses logarithms beyond n ≈ 170 (Press et al., 2007).

Is the calculator suitable for confidence intervals?

Use its probabilities to build intervals—e.g., two-tailed α = 0.05 uses z = ±1.96 (ISO 3534-1, 2006).

Why must σ and λ be positive?

Negative dispersion contradicts probability theory; “variance cannot be negative” (Montgomery, 2020).