Probability Calculator: AND, OR, and Binomial

This page covers P(A and B), P(A or B), complements, independence versus dependence, binomial PMF, and when to open combinatorics or statistics tools instead. The Probability Calculator handles AND/OR templates and binomial probabilities in the browser. Permutation Combination Calculator counts favorable and total outcomes for equally likely sample spaces. Statistics Calculator is for data summaries (mean, SD)-not the same as a single binomial coin-flip problem unless the assignment shifts to a dataset. No account.

For z-scores on measured data, see Standard deviation vs z-score. For percent language in word problems, see Percentage calculator: three kinds of percent problems.

Sample space and the basic rules

A probability is a number from 0 to 1 (or 0% to 100%) measuring likelihood.

P(certain event) = 1. P(impossible) = 0.

For equally likely finite outcomes:

P(A) = (number of outcomes in A) / (total outcomes)

Fair die: P(rolling 4) = 1/6.

Complement: P(Aᶜ) = 1 − P(A).

P(not 4 on a die) = 1 − 1/6 = 5/6.

The Probability Calculator applies these rules once you classify the event.

AND: intersection, usually multiply when independent

P(A and B) means both A and B occur (intersection).

Independent events: one does not affect the other.

P(A and B) = P(A) · P(B) when independent.

Example: fair coin twice. A = first H, B = second H.

P(A and B) = (1/2)(1/2) = 1/4.

Example: die then coin. P(6 and H) = (1/6)(1/2) = 1/12.

Dependent events: multiply conditional probabilities.

P(A and B) = P(A) · P(B|A) = P(B) · P(A|B).

Draw two aces without replacement from a deck:

P(ace then ace) = (4/52)(3/51) = 12/2652 = 1/221.

Not (4/52)²-that would assume replacement.

Tree diagrams show AND as branches along a path-multiply along the path.

OR: union, add then subtract overlap

P(A or B) means at least one of A or B (union).

General addition rule:

P(A or B) = P(A) + P(B) − P(A and B)

Subtract P(A and B) so outcomes counted in both A and B are not double-counted.

Mutually exclusive events cannot both happen: P(A and B) = 0.

Then P(A or B) = P(A) + P(B).

Example: one die roll. A = {1,2}, B = {5,6}. Disjoint.

P(A or B) = 2/6 + 2/6 = 4/6 = 2/3.

Example: one die. A = even, B = multiple of 3.

Even: {2,4,6}. Mult 3: {3,6}. Overlap {6}.

P(A) = 3/6, P(B) = 2/6, P(A and B) = 1/6.

P(A or B) = 3/6 + 2/6 − 1/6 = 4/6 = 2/3.

Complement: the "at least one" shortcut

At least one of several tries often beats direct OR expansion.

P(at least one head in 3 flips) = 1 − P(no heads) = 1 − (1/2)³ = 1 − 1/8 = 7/8.

P(at least one six in two dice) = 1 − (5/6)² = 1 − 25/36 = 11/36.

Identify "none" or "all fail" first, subtract from 1.

The probability calculator should offer complement mode or fields for "at least k successes" in binomial form.

Binomial model: n trials, k successes

Conditions for binomial:

1. Fixed number of trials n.
2. Each trial independent.
3. Two outcomes: success (p) or failure (1 − p).
4. Same p every trial.

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

C(n, k) = n! / (k!(n−k)!) is the binomial coefficient.

Example: 10 fair coin flips, exactly 3 heads. p = 1/2, n = 10, k = 3.

P = C(10,3) (1/2)³ (1/2)⁷ = 120 · (1/2)¹⁰ = 120/1024 ≈ 0.117.

P(X ≤ k) or P(X ≥ k) sums multiple binomial terms-use calculator cumulative mode or add terms by hand for small n.

Example: at least 2 heads in 3 flips (binomial n=3, p=1/2):

P(X ≥ 2) = P(2) + P(3) = C(3,2)(1/2)³ + C(3,3)(1/2)³ = 3/8 + 1/8 = 1/2.

Same as 1 − P(0 heads) − P(1 head) = 1 − 1/8 − 3/8 = 1/2.

Combinatorics when the sample space is counting

When outcomes are equally likely but not independent Bernoulli trials, count:

P = favorable / total

Committee of 3 from 10 people, none specified role: combinations.

C(10,3) = 120 total committees.

If 4 are women, favorable with exactly 2 women (example structure): C(4,2)C(6,1) / C(10,3).

The Permutation Combination Calculator computes C(n,k) and P(n,k) without memorizing large factorials.

Order matters → permutations. Order does not → combinations.

Probability of a full house in poker is not intro binomial-it is huge combinatorics. Match tool to problem scale.

Statistics calculator versus probability calculator

Probability calculator: model-based-coins, dice, binomial n and p, AND/OR with given probabilities.

Statistics calculator: data-based-mean, median, standard deviation of a list you measured.

If the problem gives raw test scores and asks for the percentage above 80, that is descriptive statistics (Statistics Calculator), not binomial unless the course models each score as a trial.

If the problem says "80% pass rate, 10 students, find P(exactly 7 pass)", that is binomial with p = 0.8, n = 10.

Homework workflow with DevOkk tools

Step 1 - Model the experiment. Independent? Replacement? Fixed n?

Step 2 - Classify language. AND, OR, complement, exactly k, at least k.

Step 3 - Pick formula. Multiply, addition rule, complement, or binomial.

Step 4 - Compute. Fraction exact or decimal as instructed.

Step 5 - Check. Probabilities sum to 1 over a partition; complement sanity.

Open Probability Calculator for AND/OR and binomial fields. Open Permutation Combination Calculator when counting outcomes. Open Statistics Calculator for dataset summaries, not coin flips.

Worked examples

AND independent: P(rain) = 0.3, P(traffic) = 0.4 independent.

P(rain and traffic) = 0.12.

OR overlapping: P(A) = 0.5, P(B) = 0.4, P(A and B) = 0.2.

P(A or B) = 0.5 + 0.4 − 0.2 = 0.7.

Complement: P(at least one success in 4 trials, p = 0.1) = 1 − (0.9)⁴ ≈ 0.344.

Binomial: n = 5, p = 0.2, k = 2.

P = C(5,2)(0.2)²(0.8)³ = 10 × 0.04 × 0.512 = 0.2048.

Conditional probability and Bayes (intro level)

When events are dependent, P(A|B) is the probability of A given B already happened.

P(A|B) = P(A and B) / P(B) when P(B) > 0.

Example: deck without replacement. P(second ace | first ace) = 3/51, not 4/52.

Bayes (some courses): reverse a conditional:

P(A|B) = P(B|A) P(A) / P(B).

Medical test screening uses this: rare disease, nonzero false positive rate-P(disease | positive test) is not the same as P(positive | disease). Set up a table of counts or probabilities before multiplying.

The Probability Calculator handles independence templates; conditional tables often stay on paper unless the page accepts P(B|A) fields.

Expected value with binomial shortcuts

For binomial X with trials n and success probability p:

E[X] = np (expected number of successes).

Var(X) = np(1 − p).

Fair coin 100 flips: E[heads] = 50, Var = 25, SD = 5.

You do not sum k·P(X=k) by hand when these formulas apply-use them to sanity-check calculator output.

For two independent binomials X and Y, E[X + Y] = E[X] + E[Y] when the story supports adding counts across separate experiments.

Venn diagrams for AND and OR

Draw A and B as overlapping circles inside a rectangle (sample space).

P(A and B) → intersection overlap.

P(A or B) → both circles combined, counted once.

P(A and Bᶜ) → part of A outside B.

Shading matches set notation on many exams. When overlap is empty (mutually exclusive), circles do not intersect and P(A or B) = P(A) + P(B) simplifies.

Translate Venn shading to numbers before opening the calculator-especially when three events appear in advanced homework.

Common traps

Adding for AND or multiplying for OR without checking.

Assuming independence when drawing without replacement.

Forgetting overlap in OR.

Wrong n or p in binomial (variable p across trials breaks the model).

Using binomial for "without replacement" from a small finite population without adjustment (hypergeometric in advanced courses).

Confusing "at least" with "exactly".

Privacy and what the calculator does not do

Probabilities you enter are processed in the browser. No account. The tool returns numeric probabilities; tree diagrams and combination setups still belong on paper when graded.

The rules you keep after the unit test

AND often multiplies when independent; otherwise condition. OR adds and subtracts overlap. Complement handles at least one. Binomial counts k successes in n trials with fixed p. Combinatorics counts when outcomes are equally likely.

Use Probability Calculator for and-or and binomial homework. Use Permutation Combination Calculator for counting favorable outcomes. Use Statistics Calculator when the input is a data list, not a model. Pair with Standard deviation vs z-score for data z-scores and Percentage calculator when answers are reported as percents. Classify the wording, pick the rule, compute, verify.