What is probability in simple terms?

Probability is a number between 0 and 1 that quantifies how likely an event is to occur. A probability of 0 means the event cannot happen, 1 means it is certain, and values like 0.25 or 0.7 describe intermediate levels of likelihood.

How do you calculate basic probability?

When all outcomes are equally likely, the probability of an event is the number of favourable outcomes divided by the total number of possible outcomes: P(A) = favourable / total. For example, rolling a 3 on a fair six-sided die has probability 1/6.

What is the difference between independent and mutually exclusive events?

Independent events do not influence one another, so P(A and B) = P(A) × P(B). Mutually exclusive events cannot happen at the same time, so P(A and B) = 0 and P(A or B) = P(A) + P(B). Many real-world examples are neither perfectly independent nor perfectly exclusive.

What is a binomial probability?

A binomial probability describes the chance of obtaining a given number of successes in a fixed number of independent trials when each trial has the same probability of success p. The probability of exactly k successes out of n is given by the binomial formula: P(X = k) = C(n, k) p^k (1 − p)^(n − k).

Can this calculator replace formal statistical software?

This calculator is designed as an educational and quick-check tool for standard probability problems. For advanced research, large datasets or specialised models you should use dedicated statistical software or programming libraries and consult a statistician when decisions have high stakes.

Probability Calculator

Single, combined & binomial

Compute basic probabilities from favourable outcomes, combine independent or mutually exclusive events, and evaluate binomial probabilities – all in one tool, with clear formulas and explanations.

Good for: homework checks, teaching, probability intuition, quick sanity checks before sending a report or implementing logic in code or spreadsheets.

Single event probability – favourable vs total outcomes

Use this mode when all outcomes are equally likely. Example: probability of drawing a red ball from an urn, rolling an even number on a fair die, or picking a specific card from a shuffled deck.

Favourable outcomes

Outcomes that count as “success”.

Total outcomes

All equally likely outcomes.

Complement (optional)

P(A) = favourable / total

P(not A) = 1 − P(A)

Fraction form

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Decimal form

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Percentage & complement

P(A): —

P(not A): —

Combined events – A and B, A or B

Compare independent and mutually exclusive events. Probabilities can be provided as decimals (0.3), percentages (30) or fractions (3/10).

P(A)

Probability of event A.

P(B)

Probability of event B.

Relationship between A and B Independent (typical coin tosses) Mutually exclusive (cannot happen together)

P(A and B)

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Independent: P(A) × P(B); exclusive: 0.

P(A or B)

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Independent: P(A)+P(B)−P(A and B); exclusive: P(A)+P(B).

Complements

P(not A) = —

P(not B) = —

Binomial probability – exactly k successes in n trials

Model repeated independent trials, such as multiple coin tosses, defect checks or click-through events. Assumes the probability of success stays constant from trial to trial.

Number of trials (n)

Max 200 for numerical stability.

Successes (k)

0 ≤ k ≤ n.

Success probability p

As a decimal between 0 and 1.

Mode

Exact or cumulative probability.

Adjust k quickly: k = 3

Probability

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Displayed as decimal and percentage.

Odds (approx.)

—

Odds in favour of the event, when defined.

Quick summary

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Formula used

For X ~ Bin(n, p), the probability of exactly k successes is

P(X = k) = C(n, k) · p^k · (1 − p)^{n − k}

where C(n, k) is the binomial coefficient “n choose k”.

Core definitions: events, outcomes and probability

In probability theory, we start from the sample space (all possible outcomes) and define an event as any collection of outcomes we care about. If all outcomes are equally likely, the probability of an event A is

P(A) = number of favourable outcomes / number of possible outcomes.

Example: rolling a 4 on a fair die: P(4) = 1/6; rolling an even number: P({2,4,6}) = 3/6 = 1/2.

Independent vs mutually exclusive events

The way we combine probabilities depends on how events are related:

Independent events do not affect each other. For independent A and B: P(A and B) = P(A) · P(B).
Mutually exclusive events cannot occur at the same time. For mutually exclusive A and B: P(A and B) = 0 and P(A or B) = P(A) + P(B).
In general, P(A or B) = P(A) + P(B) − P(A and B).

Binomial distribution in practice

The binomial model is a workhorse in statistics, quality control and A/B testing. It applies when:

You repeat the same trial a fixed number of times n,
each trial has just two outcomes (success/failure),
the probability of success p is constant, and
trials are independent.

Under these assumptions, the random variable X = number of successes follows a binomial distribution. The calculator uses this distribution to compute probabilities like:

“Exactly k successes” P(X = k),
“At most k successes” P(X ≤ k),
“At least k successes” P(X ≥ k).

FAQ: using this probability calculator correctly

Can I mix fractions, decimals and percentages?

In the combined events section you can enter probabilities as fractions (3/10), decimals (0.3) or percentages (30) – the calculator normalises them internally as decimal probabilities between 0 and 1. For the binomial section, use a decimal p between 0 and 1.

What if my trials are not independent?

If trials influence each other (for example, drawing cards without replacement), the binomial model is only an approximation. For small samples this can make a noticeable difference, so prefer exact combinatorial calculations or specialised models (such as the hypergeometric distribution).

Can I use this for real-world risk assessment?

You can use this calculator to clarify assumptions and do order-of-magnitude checks. However, real risk assessment often requires more complex models, empirical data and domain expertise. For high-impact decisions, always review your reasoning with a statistician or subject-matter expert.