AI Grounds

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Guided discovery

Bernoulli, Categorical & Binomial

Match the probability model to the question.

Did one event succeed?

One trial: 0 means failure, 1 means success.

Chance of success on each trial. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.

Probability mass

Mass means the chance assigned to a possible outcome. The vertical scale stays at 0–100%; outlined bars are most likely, including ties. Tiny probabilities have tiny bars; they are not boosted to a minimum height.

100%50%0%
38%
0
62%
1

Scroll the bars horizontally if needed. The complete probability table below provides the same data.

Mean

Probability-weighted average of numerical outcomes.

0.62

0 × (1 − p) + 1 × p = 0.62

The mean need not be a possible single outcome.

Variance

Spread of the coded 0/1 outcome.

0.2356

p × (1 − p)

Most likely

Outcome(s) with the largest probability.

1

Mode: a tie outlines every equal maximum.

Formula

P(X = x) = p^x(1 - p)^(1 - x)

P(X = 1) = p, P(X = 0) = 1 - p

Complete probability table
Probability percentages may be rounded; full-precision masses sum to one. Small positive probabilities below 0.01% are explicitly marked.
OutcomeProbabilityMost likely?
failure38%No
success62%Yes
Total100%

Bernoulli asks about one yes/no outcome. Categorical asks which class label occurs. Binomial asks how many successes occur in independent equal-p trials. These are theoretical probabilities, not sampled frequencies.

Bernoulli Trial, p 62%. Most likely: 1. Mean 0.62; variance 0.2356. Total mass 100%.