Data Source and Methodology
Authoritative reference: NIST/SEMATECH e‑Handbook of Statistical Methods (2012). National Institute of Standards and Technology. Sections on Probability, Binomial Distribution, and Bayes’ theorem. Access at: https://www.itl.nist.gov/div898/handbook/. All calculations are strictly based on the formulas and data provided by this source.
The Formula Explained
Simple probability: $$P(E) = \frac{\text{favorable}}{\text{total}}$$
Complement: $$P(E^{c}) = 1 - P(E)$$
Independent events: $$P(A \cap B) = P(A)\,P(B), \quad P(A \cup B) = P(A) + P(B) - P(A)P(B)$$
At least once in n trials: $$P(\text{at least one}) = 1 - (1 - p)^{n}$$
Binomial (X ∼ Bin(n,p)): $$P(X=k)=\binom{n}{k}\,p^{k}(1-p)^{n-k}$$
Cumulative: $$P(X \le k)=\sum_{i=0}^{k}\binom{n}{i}p^{i}(1-p)^{n-i}, \quad P(X \ge k)=1-P(X \le k-1)$$
Bayes’ theorem: $$P(A \mid B)=\frac{P(B \mid A)\,P(A)}{P(B \mid A)\,P(A)+P(B \mid \neg A)\,[1-P(A)]}$$
Glossary of Variables
| Symbol/Field | Meaning | Valid range |
|---|---|---|
| p | Probability of success (single trial or event) | 0 ≤ p ≤ 1 |
| n | Number of independent trials | n ∈ {0,1,2,…} |
| k | Number of successes | 0 ≤ k ≤ n |
| Favorable | Count of favorable outcomes | ≥ 0 integer |
| Total | Total equally likely outcomes | ≥ 1 integer |
| P(A), P(B) | Probabilities of events A and B | 0 ≤ P ≤ 1 |
| P(B|A), P(B|¬A) | Conditional probabilities for Bayes’ theorem | 0 ≤ P ≤ 1 |
| Probability | Result in [0,1] | — |
| Percentage | Probability × 100% | — |
| Odds (in favor) | p / (1 − p) | Defined for 0<p<1 |
All inputs are validated inline with clear guidance. Use decimals for probabilities (e.g., 0.2 = 20%).
Worked Example
How It Works: A Step‑by‑Step Example
Question: What is the probability of getting at least 3 heads in 5 tosses of a fair coin?
- Identify the model: X ∼ Bin(n=5, p=0.5). We want P(X ≥ 3).
- Use the cumulative binomial formula: $$P(X \ge 3) = 1 - P(X \le 2) = \sum_{k=3}^{5}\binom{5}{k} (0.5)^k (0.5)^{5-k}$$
- Compute terms: C(5,3)=10, C(5,4)=5, C(5,5)=1. Since (0.5)^5 = 1/32: $$P = (10+5+1)\cdot \frac{1}{32} = \frac{16}{32} = 0.5 = 50\%.$$
In the calculator: choose “Binomial distribution”, set n=5, k=3, p=0.5, and operator “≥”. The output will show 0.5 (50%) and odds 1:1.
Frequently Asked Questions (FAQ)
Do I enter probabilities as decimals or percentages?
Enter decimals between 0 and 1. To convert a percent to a decimal, divide by 100 (e.g., 25% → 0.25).
What if events are not independent?
Use Bayes’ theorem or compute conditional probabilities. The AND/OR formulas shown here assume independence; if not independent, use P(A ∩ B)=P(A)P(B|A).
How accurate is the binomial computation?
Exact probabilities use stable algorithms. Cumulative values use an efficient recursive method that avoids overflow for typical ranges (n ≤ 10,000).
What are “odds in favor”?
They are p/(1−p). For p=0.75, odds are 3:1 in favor. For p=0.2, odds are 1:4 (against).
Can the result be exactly 0 or 1?
Yes. For impossible events p=0, and for certain events p=1. The calculator handles these edge cases explicitly.
When should I prefer “At least once” over Binomial?
If you only need P(at least one success) across n identical trials, the “At least once” mode is simpler: 1 − (1 − p)^n.
Which sources back these formulas?
The tool follows the NIST/SEMATECH e‑Handbook of Statistical Methods and standard probability texts such as Ross (A First Course in Probability).