Probability Calculator

Understanding Probability Basics

Probability measures how likely an event is to happen, expressed as a number between 0 and 1. A probability of 1 means the event is certain, while 0 means it’s impossible. The higher the probability, the more likely the event will occur.

To calculate probability, divide the number of desired outcomes by the total possible outcomes. However, probabilities can change based on whether events are independent, mutually exclusive, or conditional. Our probability calculator helps you determine:

• The chance that neither Event A nor B occurs

• The probability that A or B happens (when they can occur together)

• The likelihood that both A and B occur

• The probability that either A or B happens, but not both

Calculating the Complement (Opposite) of an Event

If you know the probability of an event (P(A)), finding its complement (P(A’)) is simple:

P(A’) = 1 – P(A)

Example:

• If there’s a 65% chance Bob won’t do his homework (P(A) = 0.65),

• Then, the probability he will do it is:
  P(A’) = 1 – 0.65 = 0.35 (or 35%)

This applies to any event. Note that P(A) and P(B) can be different—they don’t have to add up to 1.

Probability of Both Events Happening (Intersection)

The intersection (P(A ∩ B)) is the chance that both A and B occur. If A and B are independent (one doesn’t affect the other), multiply their probabilities:

P(A ∩ B) = P(A) × P(B)

Example:

• Rolling a die twice:

  • Probability of first roll being 6 = 1/6

  • Probability of second roll being 6 = 1/6

  • Combined probability = (1/6) × (1/6) = 1/36

For dependent events (where one affects the other), use conditional probability (P(B|A)):

Example:

• A bag has 7 black and 3 blue marbles.

• Probability of first drawing a blue marble (P(A)) = 3/10

• If a blue marble is removed, the chance of then drawing a black marble (P(B|A)) = 7/9

• Combined probability = (3/10) × (7/9) ≈ 0.233 (23.3%)

Probability of Either Event Happening (Union)

The union (P(A ∪ B)) calculates the chance that A or B (or both) occur.

• Mutually exclusive events (can’t happen together):
  P(A ∪ B) = P(A) + P(B)
  (Example: Rolling an odd or even number on a die—they can’t both happen.)

• Non-mutually exclusive events (can happen together):
  P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Example:

• Rolling a die:

  • Probability of an even number (2,4,6) = 3/6

  • Probability of a multiple of 3 (3,6) = 2/6

  • Overlap (6) = 1/6

  • Total probability = (3/6) + (2/6) – (1/6) = 4/6 ≈ 66.67%

Exclusive OR (One or the Other, But Not Both)

The “Exclusive OR” (P(A XOR B)) calculates the probability that either A or B occurs, but not both:

P(A XOR B) = P(A) + P(B) – 2 × P(A ∩ B)

Example:

• Halloween candy choice:

  • P(Reese’s) = 0.65

  • P(Snickers) = 0.349

  • P(Both) = 0 (since the child must pick only one)

  • Probability of one or the other = 0.65 + 0.349 – 0 = 0.999 (99.9%)

Normal Distribution (Bell Curve) Probability

The normal distribution predicts probabilities for continuous data (e.g., heights, test scores). It follows a bell curve where:

• μ (mean) = Average value

• σ (standard deviation) = Measure of spread

Example:

• Male student heights:

  • Mean (μ) = 68 inches

  • Standard deviation (σ) = 4 inches

• Probability a student is between 60 and 72 inches tall:

  • Convert to Z-scores:

    • (60 – 68)/4 = –2

    • (72 – 68)/4 = 1

  • Check a Z-table:

    • P(0 to 2) = 0.47725

    • P(0 to 1) = 0.34134

    • Total probability = 0.47725 + 0.34134 ≈ 81.86%

Z-Table Reference (Standard Normal Distribution)