Probability Calculator

What is Probability?

Probability measures how likely an event is to occur, expressed as a number between 0 and 1:

• 0 = Impossible (will never happen)
• 1 = Certain (will definitely happen)
• 0.5 = Equally likely to happen or not happen

Basic Probability Formula:

P(Event) = Number of Favorable Outcomes / Total Number of Possible Outcomes

Example: Rolling a 6 on a standard die

• Favorable outcomes: 1 (only one face shows 6)
• Total outcomes: 6 (six faces total)
• P(6) = 1/6 = 0.167 = 16.7%

Probability Representations:

Converting Between Formats:

• Decimal to Percentage: Multiply by 100 (0.25 x 100 = 25%)
• Percentage to Decimal: Divide by 100 (25% / 100 = 0.25)
• Decimal to Odds: If P = 0.25, odds for = 0.25 : 0.75 = 1:3
• Odds to Probability: If odds are 1:3, P = 1/(1+3) = 0.25

1. Complementary Rule (NOT)

The probability of an event NOT occurring equals 1 minus the probability it does occur:

P(NOT A) = 1 - P(A)

Example: If there's a 30% chance of rain, there's a 70% chance of no rain.

• P(Rain) = 0.30
• P(No Rain) = 1 - 0.30 = 0.70 = 70%

2. Multiplication Rule (AND) - Independent Events

For independent events (where one doesn't affect the other), multiply the probabilities:

P(A AND B) = P(A) x P(B)

Example: Probability of flipping heads twice in a row:

• P(Heads) = 0.5
• P(Heads AND Heads) = 0.5 x 0.5 = 0.25 = 25%

3. Addition Rule (OR) - Mutually Exclusive Events

For events that cannot happen simultaneously:

P(A OR B) = P(A) + P(B)

Example: Probability of rolling a 1 or 6 on a die:

• P(1) = 1/6, P(6) = 1/6
• P(1 OR 6) = 1/6 + 1/6 = 2/6 = 1/3 = 33.3%

For Non-Mutually Exclusive Events:

P(A OR B) = P(A) + P(B) - P(A AND B)

This subtraction prevents double-counting outcomes that satisfy both A and B.

Independent Events: One event has no effect on the probability of another.

Examples of Independent Events:

• Coin flips (each flip is independent)
• Dice rolls (each roll is independent)
• Separate lottery drawings
• Weather on different, distant days
• Manufacturing defects on different production lines

Key Property: P(A AND B) = P(A) x P(B)

Dependent Events: The outcome of one event affects the probability of another.

Examples of Dependent Events:

• Drawing cards without replacement
• Selecting items from a bag without returning them
• Chain reactions in chemistry
• Conditional medical diagnoses

Key Property: P(A AND B) = P(A) x P(B|A)

Where P(B|A) is "probability of B given A occurred."

Example: Drawing Two Aces (No Replacement)

• P(First Ace) = 4/52
• P(Second Ace | First Ace) = 3/51 (only 3 aces left, 51 cards total)
• P(Two Aces) = (4/52) x (3/51) = 12/2652 = 1/221 = 0.45%

Example: Drawing Two Aces (With Replacement)

• P(First Ace) = 4/52
• P(Second Ace) = 4/52 (card returned, deck is full again)
• P(Two Aces) = (4/52) x (4/52) = 16/2704 = 1/169 = 0.59%

Notice that replacement makes the probability higher because aces remain available.

The "At Least One" Problem:

When calculating the probability of at least one success in multiple trials, use the complement:

P(At Least One) = 1 - P(None)

Why This Works: "At least one" means 1, 2, 3, or more. Instead of adding all these, calculate the opposite (none) and subtract from 1.

Example: At Least One Head in 3 Coin Flips

Direct approach (tedious):

• P(Exactly 1 head) + P(Exactly 2 heads) + P(Exactly 3 heads)

Complement approach (easy):

• P(No heads) = P(T,T,T) = 0.5 x 0.5 x 0.5 = 0.125
• P(At least 1 head) = 1 - 0.125 = 0.875 = 87.5%

Example: At Least One 6 in 4 Dice Rolls

• P(Not 6) = 5/6
• P(No 6s in 4 rolls) = (5/6)^4 = 625/1296 = 0.482
• P(At least one 6) = 1 - 0.482 = 0.518 = 51.8%

General Formula for n Independent Trials:

P(At Least One Success) = 1 - (1 - P)^n

Where P is the probability of success on a single trial and n is the number of trials.

Quick Reference:

1. Theoretical (Classical) Probability

Based on known possible outcomes, assuming all are equally likely.

P = Favorable Outcomes / Total Possible Outcomes

Examples:

• Fair coin: P(Heads) = 1/2
• Fair die: P(3) = 1/6
• Deck of cards: P(Ace) = 4/52 = 1/13

2. Experimental (Empirical) Probability

Based on observed data from experiments or historical records.

P = Number of Times Event Occurred / Total Number of Trials

Examples:

• Free throw percentage = Makes / Attempts
• Batting average = Hits / At-bats
• Quality control: Defects / Units inspected

3. Subjective Probability

Based on personal judgment, experience, or beliefs when data is unavailable.

Examples:

• "I think there's a 70% chance the project finishes on time"
• Expert opinions on unprecedented events
• Risk assessment for new ventures

4. Conditional Probability

The probability of an event given that another event has occurred.

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

Example: P(drawing a heart | drawing a red card)

• P(Heart AND Red) = 13/52 (all hearts are red)
• P(Red) = 26/52
• P(Heart|Red) = (13/52) / (26/52) = 13/26 = 1/2 = 50%

Understanding Odds:

Odds and probability express the same concept differently:

• Probability: Successes / Total outcomes
• Odds For: Successes : Failures
• Odds Against: Failures : Successes

Converting Probability to Odds:

If P(event) = 0.25 (25%):

• Successes = 0.25 (or 1 part)
• Failures = 0.75 (or 3 parts)
• Odds For = 1:3 ("1 to 3")
• Odds Against = 3:1 ("3 to 1")

Converting Odds to Probability:

If odds for are A:B:

P = A / (A + B)

Example: Odds of 2:5

• P = 2 / (2 + 5) = 2/7 = 28.6%

Quick Reference Table:

Betting Odds Formats:

• Decimal Odds (Europe): 4.00 means $4 return per $1 bet (includes stake)
• Fractional Odds (UK): 3/1 means $3 profit per $1 bet
• American Odds (US): +300 means $300 profit per $100 bet

Use our Betting Odds Calculator for detailed conversions.

Common Probability Distributions:

1. Uniform Distribution All outcomes equally likely.

• Example: Fair die, each face has P = 1/6
• Example: Random number 1-100, each has P = 1/100

2. Binomial Distribution Number of successes in n independent trials, each with probability p.

P(k successes) = C(n,k) x p^k x (1-p)^(n-k)

Where C(n,k) = combinations of n items taken k at a time.

Example: P(exactly 3 heads in 5 coin flips)

• n = 5, k = 3, p = 0.5
• C(5,3) = 10
• P = 10 x 0.5^3 x 0.5^2 = 10 x 0.125 x 0.25 = 0.3125 = 31.25%

3. Normal Distribution (Bell Curve) Continuous distribution where most values cluster around the mean.

• 68% within 1 standard deviation
• 95% within 2 standard deviations
• 99.7% within 3 standard deviations

4. Poisson Distribution Number of events in a fixed interval when events occur independently at a constant rate.

• Example: Number of calls to a call center per hour
• Example: Number of typos per page

Expected Value: The weighted average of all possible outcomes:

E(X) = Sum of (outcome x probability)

Use our Expected Value Calculator for detailed calculations.

Medical Testing: Understanding false positives and false negatives requires probability:

• Sensitivity: P(Positive Test | Disease)
• Specificity: P(Negative Test | No Disease)
• Predictive Value: P(Disease | Positive Test)

Weather Forecasting: "30% chance of rain" means: Out of many days with similar conditions, about 30% experienced rain.

Insurance: Insurers calculate premiums based on:

• P(claim) x Average claim amount = Expected payout
• Premium > Expected payout for profitability

Quality Control: Manufacturing uses probability for:

• Acceptance sampling: P(batch acceptable | sample results)
• Six Sigma: P(defect) < 3.4 per million

Sports Analytics:

• Win probability models
• Player performance projections
• Game strategy optimization

Finance:

• Risk assessment (Value at Risk)
• Option pricing models
• Portfolio optimization

Gaming:

• Expected value of bets
• Optimal strategy calculations
• House edge determination

Project Management:

• PERT estimates using probability distributions
• Risk quantification
• Schedule probability analysis

1. Gambler's Fallacy Believing past independent events affect future outcomes.

Wrong: "The roulette wheel landed on red 5 times, so black is due!" Reality: Each spin is independent. P(Black) = 18/37 every single time.

2. Ignoring Base Rates Not considering how common something is overall.

Example: A test is 99% accurate. You test positive. What's P(disease)? It depends on how common the disease is! If 1 in 10,000 have it, most positive tests are false positives.

3. Conjunction Fallacy Believing specific scenarios are more likely than general ones.

Wrong: P(A and B) > P(A) Reality: Adding conditions can only reduce or maintain probability, never increase it.

4. Availability Heuristic Overestimating probability of memorable events.

• Shark attacks seem common (memorable news stories)
• Reality: ~5 deaths/year worldwide vs. 1.35 million car deaths

5. Confusing Conditional Probabilities P(A|B) is NOT the same as P(B|A).

Example:

• P(Rain | Clouds) might be 60%
• P(Clouds | Rain) is essentially 100%

6. Multiplication Without Independence Only multiply directly for independent events.

Wrong: P(both aces) = (4/52) x (4/52) when drawing without replacement Right: P(both aces) = (4/52) x (3/51) accounting for the first card drawn

Basic Formulas:

Multiple Independent Events:

Special Calculations:

Odds Conversions: