Understanding Probability

Probability measures how likely an event is to occur. It ranges from 0 (impossible) to 1 (certain). Multiply by 100 to express as a percentage.

P(A) = Favorable outcomes / Total outcomes
P(A and B) = P(A) × P(B) [if independent]
P(A or B) = P(A) + P(B) − P(A and B)

Probability Examples

Flipping a coin: P(Heads) = 1/2 = 0.5 = 50%
Rolling a 6: P(6) = 1/6 ≈ 0.167 = 16.7%
Drawing a red card: P(Red) = 26/52 = 0.5 = 50%

Probability Rules — Complete Reference

Rule	Formula	Example
Basic probability	P(A) = favourable/total	Die: P(6) = 1/6
Complement	P(not A) = 1 − P(A)	P(not 6) = 5/6
AND (independent)	P(A and B) = P(A) × P(B)	P(HH) = 0.5×0.5=0.25
OR (mutually exclusive)	P(A or B) = P(A) + P(B)	P(1 or 2 on die) = 2/6
OR (not exclusive)	P(A or B) = P(A)+P(B)−P(A∩B)	General addition rule
Conditional	P(A\|B) = P(A∩B)/P(B)	Given B occurred, P of A

Common Probability Examples

Fair coin: P(H)=0.5, P(T)=0.5
Fair die: P(any face)=1/6 ≈ 0.167
Deck of cards: P(Ace)=4/52≈0.077, P(Red)=26/52=0.5
Two dice sum=7: 6/36=1/6≈0.167
Lottery 6/49: C(49,6)⁻¹ ≈ 1 in 13.98 million

Expected Value

Expected value E(X) is the average result you'd get over many trials:

E(X) = Σ [x × P(x)]
Example: Die roll E = (1+2+3+4+5+6)/6 = 3.5
You "expect" 3.5 on average per roll.

Real-World Applications

Insurance: Actuaries calculate premiums using probability of claims
Medicine: Clinical trial success rates, disease prevalence, diagnostic test accuracy
Finance: Option pricing (Black-Scholes model uses probability)
Weather forecasting: "70% chance of rain" is a probability
Games: Casino odds, card game strategy, lottery expected value

Probability Calculator