How do you calculate basic probability?

Basic probability is P(A) = favourable outcomes / total outcomes. Every outcome must be equally likely. Example: rolling a 4 on a fair die. - Favourable: 1 - Total: 6 - P(rolling 4) = 1/6 = 0.1667 Complement rule: P(A') = 1 - P(A). If P(rolling 4) = 1/6, then P(not 4) = 5/6. Odds FOR = favourable : unfavourable (1:5 for rolling a 4). Odds AGAINST = unfavourable : favourable (5:1). To convert odds to probability: P = favourable / (favourable + unfavourable).

What is the difference between P(A or B) and P(A and B)?

P(A or B) is the union — at least one of A or B occurs. P(A and B) is the intersection — both occur simultaneously. P(AuB) = P(A) + P(B) - P(AnB) The subtraction removes double-counting where both happen. For P(A)=0.3, P(B)=0.4, P(AnB)=0.12: P(AuB) = 0.3 + 0.4 - 0.12 = 0.58 Special cases: - Mutually exclusive: P(AnB)=0, so P(AuB) = P(A)+P(B) - Independent: P(AnB) = P(A)xP(B). Knowing A tells nothing about B. - Test: if P(AnB) = P(A)xP(B) the events are independent; otherwise dependent.

What is conditional probability and how is it calculated?

Conditional probability P(A|B) is the probability that A occurs given B has already occurred. It narrows the sample space to outcomes where B is true. Formula: P(A|B) = P(AnB) / P(B) Example: a bag has 10 balls: 4 red, 3 blue, 3 green. A ball drawn is not green (B). - P(AnB) = P(red) = 4/10 = 0.4 - P(B) = 7/10 = 0.7 - P(A|B) = 0.4/0.7 = 0.571 Key points: - P(A|B) is not equal to P(B|A) in general. - If independent: P(A|B) = P(A). - Bayes theorem: P(A|B) = P(B|A) x P(A) / P(B).

How does the binomial distribution work?

The binomial distribution models the number of successes in n independent trials each with probability p. Requirements: - Fixed number of trials n - Each trial independent - Two outcomes per trial: success (p) or failure (1-p) - Same p for every trial For n=10, p=0.3, k=3: - C(10,3) = 120 - P(X=3) = 120 x 0.3^3 x 0.7^7 = 120 x 0.027 x 0.0823543 = 0.26683 - Mean = np = 3 - Variance = np(1-p) = 2.1 Real-world uses: defective items in a batch, heads in coin flips, customers who purchase.

What is the difference between probability and odds?

Both measure likelihood but express it differently. Probability: P(A) = favourable / total. Always between 0 and 1. Odds for: favourable : unfavourable. E.g. 3:7. Odds against: unfavourable : favourable. E.g. 7:3. Converting: - Probability to odds: P/(1-P). P=0.3 gives odds 0.3/0.7 = 3:7. - Odds to probability: favourable/(favourable+unfavourable). 3:7 gives P=3/10=0.3. Odds are common in sports betting, horse racing, and medical studies (odds ratio). Probability is more intuitive for calculation.

Probability Calculator — Basic, Combined, Conditional and Binomial

The Probability Calculator covers four types of probability in one tool. Basic Probability computes P(A), the complement, and odds from favourable and total outcomes. Combined Events finds the union, intersection, conditional probabilities, and checks independence with a live Venn diagram. Condition...

PROBABILITY CALCULATOR

Favourable outcomes

e.g. 3 red balls

Total outcomes

e.g. 10 balls total

PROBABILITY

0.3

= 3/10 = 30%

P(A'): complement

0.7 (70%)

Odds for

3 : 7

Odds against

7 : 3

Check P+P'

0.3 + 0.7 = 1 ✓

VENN DIAGRAM

PROBABILITY

30%

P(A) = 0.3P(A') = 0.7

STEP BY STEP

P(A) = Favourable / Total

P(A) = 3 / 10

P(A) = 0.3

P(A') = 1 − P(A) = 1 − 0.3 = 0.7

Odds for A = 3 : 7

Odds against A = 7 : 3

Created with❤️byeaglecalculator.com

HOW TO USE

1
Select a mode: Basic Probability for simple events, Combined Events for P(AuB) and P(AnB), Conditional for P(A|B), or Binomial for repeated independent trials.
2
Basic mode: enter the number of favourable outcomes and total outcomes. The probability, complement, and odds appear instantly.
3
Combined mode: enter P(A), P(B), and P(AnB). The union P(AuB), conditional probabilities P(A|B) and P(B|A), and independence check are shown with a live Venn diagram.
4
Conditional mode: enter the joint probability P(AnB) and the condition P(B). The formula P(A|B) = P(AnB)/P(B) is shown step by step.
5
Binomial mode: enter n (trials), k (target successes), and p (probability per trial). P(X=k), P(X<=k), P(X>=k), mean, and variance appear with a bar chart of the full distribution.

WORKED EXAMPLE

Basic: bag has 10 balls, 3 red. P(red)=3/10=0.3. Complement=0.7. Odds=3:7. Combined: P(A)=0.3, P(B)=0.4, P(AnB)=0.12. Union=0.58. P(A|B)=0.3. Independent since 0.12=0.3x0.4. Binomial: n=10, p=0.3, k=3. C(10,3)=120. P(X=3)=120x0.027x0.0823543=0.26683. Mean=3. Variance=2.1.

FORMULAS

PROBABILITY FORMULAS

NAME	FORMULA	DESCRIPTION
Basic probability	P(A) = favourable / total	Ratio of favourable outcomes to all outcomes
Complement	P(A') = 1 - P(A)	Probability that A does NOT occur
Union (OR)	P(AuB) = P(A) + P(B) - P(AnB)	Probability that A or B or both occur
Intersection (AND)	P(AnB) = P(A) x P(B) if independent	Probability that both A and B occur
Conditional	P(A\|B) = P(AnB) / P(B)	Probability of A given that B has occurred
Independence test	P(AnB) = P(A) x P(B)	A and B are independent if and only if this holds
Binomial PMF	P(X=k) = C(n,k) x p^k x (1-p)^(n-k)	Probability of exactly k successes in n trials
Binomial CDF	P(X<=k) = sum P(X=i) for i=0 to k	Probability of at most k successes
Binomial mean	mean = np	Expected number of successes
Binomial variance	variance = np(1-p)	Variance of the binomial distribution
Combinations	C(n,k) = n! / (k! x (n-k)!)	Number of ways to choose k from n

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.