Probability Calculator

Probability measures the mathematical likelihood of an event occurring. It is represented as a real number between 0 (indicating the event is impossible) and 1 (indicating absolute certainty). In its simplest terms, probability is the number of desired outcomes divided by the total number of possible outcomes. Solving these values depends on whether the events are independent, dependent, or mutually exclusive.

Our free Probability Calculator solves multiple scenarios simultaneously. You can compute the union, intersection, complement, and exclusive OR (XOR) for two independent events, calculate probability series repetitions, or solve standard normal distribution Z-scores.

1. Probability of Two Events

When analyzing two events, A and B, with known individual probabilities P(A) and P(B), we calculate several key relational outcomes:

A. Complement of an Event (P(A’) and P(B’))

The complement represents the probability that the event does not occur:

P(A') = 1 - P(A)

The Homework Prediction: If there is a P(A) = 0.65 chance that Bob does not do his homework, his teacher Sally calculates the complement (the probability that Bob does his homework) as:

P(A') = 1 - 0.65 = 0.35 (a 35% chance).

B. Intersection of Independent Events (P(A ∩ B))

The intersection P(A ∩ B) or P(A AND B) is the joint probability that both events occur. For independent events:

P(A ∩ B) = P(A) × P(B)

Rolling Dice: The probability of rolling a 6 on two successive rolls of a single die is:

P(A ∩ B) = 1/6 × 1/6 ≈ 0.0278 (2.78%).

If events are mutually exclusive (cannot happen at the same time, like rolling a 4 and 6 on a single die roll), their intersection is zero: P(A ∩ B) = 0.

C. Intersection of Dependent Events (Conditional Probability)

If the outcome of one event depends on a previous event, we use conditional probability P(B|A)—the probability of B given that A has occurred:

P(A ∩ B) = P(A) × P(B|A)

The Marble Draw: A bag contains 10 marbles (7 black, 3 blue). We draw two marbles without replacement:

• Probability of drawing blue first: P(A) = 3/10
• Probability of drawing black second: P(B) = 7/10
• Probability of drawing black given that blue was already removed (leaving 9 total marbles): P(B|A) = 7/9
• Joint probability of blue then black: P(A ∩ B) = (3/10) × (7/9) ≈ 0.2333 (23.33%)

D. Union of Events (P(A ∪ B))

The union P(A ∪ B) or P(A OR B) is the probability that at least one of the events occurs (inclusive OR).

• For Mutually Exclusive Events: P(A ∪ B) = P(A) + P(B)
• For Non-Mutually Exclusive Events: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Rolling Dice Example: Find the probability of rolling an even number (2, 4, 6) or a multiple of 3 (3, 6):

P(A) = 3/6, P(B) = 2/6, and their intersection is the number 6: P(A ∩ B) = 1/6.

P(A ∪ B) = 3/6 + 2/6 - 1/6 = 4/6 = 2/3 (66.67%).

E. Exclusive OR (P(A Δ B))

The probability that event A or event B occurs, but not both simultaneously:

P(A Δ B) = P(A) + P(B) - 2 × P(A ∩ B)

The Halloween Candy: Children can take Snickers P(B) = 0.349 or Reese’s P(A) = 0.65, but not both. For independent choices:

P(A Δ B) = 0.65 + 0.349 - 2 × (0.65 × 0.349) = 0.999 - 0.4537 = 0.5453 (54.53%).

2. Normal (Gaussian) Distribution and Z-Scores

The normal distribution is a continuous probability distribution that clusters symmetrically around a central mean (μ) with a spread defined by the standard deviation (σ). When standardized so that μ = 0 and σ = 1, it is called the standard normal distribution.

To find the probability of a value falling between two bounds, we convert the raw bounds into standard Z-scores:

Z = (X - μ) / σ

Normal Distribution Example

Find the probability that a male college student has a height between 60 and 72 inches, given a mean of 68 inches and a standard deviation of 4 inches:

• Standardize 60 inches: Z1 = (60 - 68) / 4 = -2
• Standardize 72 inches: Z2 = (72 - 68) / 4 = 1
• From the Z-table below:
  • Area between 0 and 2 is 0.47725
  • Area between 0 and 1 is 0.34134
• Total Probability (summing the two halves): 0.47725 + 0.34134 = 0.81859 (81.859%).

Standard Normal Z-Table (0 to Z)

This table provides the cumulative area under the normal curve from the mean (0) to the standardized value Z:

Solve data spreads using our Standard Deviation Calculator or determine required parameters on the Sample Size Calculator.

Frequently Asked Questions (FAQ)

What is the difference between independent and dependent events?

Independent events have no impact on one another; the outcome of Event A does not alter the probability of Event B (e.g., flipping coins). Dependent events are linked; the outcome of Event A alters the probability of Event B (e.g., drawing cards from a deck without replacement).

How does exclusive OR differ from inclusive OR?

Inclusive OR (union, P(A ∪ B)) is satisfied if Event A occurs, Event B occurs, or both occur simultaneously. Exclusive OR (XOR, P(A Δ B)) is satisfied if Event A occurs or Event B occurs, but fails if both events occur at the same time.

Why are Z-score probabilities symmetrical?

Because the normal distribution curve is a perfectly symmetrical bell shape centered around the mean. Consequently, the area from 0 to -Z is exactly equal to the area from 0 to +Z. Z-tables leverage this property to only publish positive values, saving space.

What is a standard normal distribution?

A standard normal distribution is a special case of the normal distribution where the mean (μ) is exactly 0 and the standard deviation (σ) is exactly 1. Standardizing datasets to this format allows researchers to compare different normal variables using a single Z-table.