Calculator Inputs
Example Data Table
| Row | Evidence (E) | Observed Outcome | P(YES) Example | Notes |
|---|---|---|---|---|
| 1 | High engagement | YES | 0.72 | Frequency estimate with α=1 |
| 2 | Medium engagement | NO | 0.46 | Posterior from prior + likelihoods |
| 3 | Low engagement | NO | 0.18 | Logistic score with negative z |
| 4 | High intent signal | YES | 0.83 | Odds increase after evidence |
| 5 | Weak intent signal | NO | 0.33 | Threshold decides final label |
Formula Used
- Frequency with Laplace smoothing: P(YES) = (yes + α) / (yes + no + 2α)
- Bayes posterior: P(YES|E) = P(YES)P(E|YES) / [P(YES)P(E|YES) + P(NO)P(E|NO)]
- Logistic scoring: z = b0 + Σ(bi·xi), then P(YES) = 1 / (1 + e-z)
- Odds and log-odds: odds = p/(1-p), logit = ln(p/(1-p))
How to Use This Calculator
- Select a method that matches your data source (counts, Bayes, or logistic).
- Enter your inputs and set a decision threshold for YES labeling.
- Click Calculate Probability to show results above the form.
- Use Download CSV for spreadsheets or Download PDF for reports.
- Adjust inputs to compare scenarios and track how evidence shifts odds.
Binary outcome framing
This calculator models a two-class decision where the target is YES or NO. In analytics pipelines, these labels often represent approval, conversion, churn, fraud, or pass/fail. A binary view simplifies monitoring: you can track the base rate, the predicted YES share, and the gap between them to spot drift quickly. Teams commonly review these rates weekly, alongside sample size, because a stable probability estimate needs enough observations. Segment by region or cohort to detect bias and seasonal changes early.
Frequency estimates with smoothing
When you only have historical counts, the frequency method provides an interpretable estimate. Laplace smoothing adds α pseudo-observations to both classes, reducing extreme probabilities at small sample sizes. For example, with 1 YES and 0 NO, unsmoothed P(YES)=1.00, but with α=1 the estimate becomes (1+1)/(1+0+2)=0.67. With 2 YES and 0 NO, α=1 yields 0.75, while α=0.5 yields 0.83, showing how smoothing strength controls conservatism.
Bayesian posterior updates
Bayes mode combines a prior belief with evidence likelihoods. If the prior is 0.40 and the evidence is more compatible with YES than NO (P(E|YES)=0.80, P(E|NO)=0.20), the posterior becomes 0.73. In odds form, posterior odds equal prior odds multiplied by the likelihood ratio (0.80/0.20=4). This helps analysts explain why one strong signal can outweigh a moderate prior.
Logistic scoring and calibration
Logistic mode converts a linear score z into a probability through the sigmoid curve. Each feature contributes bi·xi, so effect size is easy to audit. Because probabilities are bounded, large positive z quickly approaches 1.00. If z=0, P(YES)=0.50; if z=2, P(YES)=0.88; if z=-2, P(YES)=0.12. In practice, you validate probabilities with calibration checks such as reliability curves or a Brier score, not only ranking metrics like AUC.
Decision thresholds and reporting
A probability becomes an action when you choose a threshold. A stricter threshold like 0.70 reduces false YES decisions but may increase missed positives. Many teams select thresholds by optimizing precision, recall, or F1, and by estimating the cost per error. The calculator reports odds and log-odds for clearer communication: odds of 3.0 mean YES is three times as likely as NO. Exporting CSV and PDF supports audits, model reviews, and stakeholder updates with consistent numbers.
FAQs
1) What does P(YES) represent?
It is the estimated probability that the outcome is YES given your selected method and inputs. It ranges from 0 to 1 and complements P(NO)=1−P(YES).
2) When should I use Laplace smoothing?
Use smoothing when counts are small or one class is missing. α prevents probabilities of exactly 0 or 1, which improves stability for reporting and downstream scoring.
3) How do I choose the prior and likelihoods in Bayes mode?
Set the prior from long-run prevalence, such as a recent YES rate. Likelihoods should capture how often evidence occurs under YES versus NO, preferably estimated from labeled samples or validated assumptions.
4) What do odds and log-odds mean?
Odds convert probability into a ratio p/(1−p). Odds of 3 mean YES is three times as likely as NO. Log-odds apply the natural log, making additive changes easier to interpret.
5) Why can logistic probabilities be off even when ranking is good?
A model can rank cases well but still output biased probabilities if data is imbalanced, features drift, or regularization is strong. Calibration checks and monitoring help align predictions with observed frequencies.
6) Which threshold should I choose?
Choose a threshold based on error costs and target metrics. Higher thresholds increase precision, lower thresholds increase recall. Start at 0.50, then test 0.70 for stricter decisions or 0.35 for broader coverage.