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Probability Calculator

Calculate probabilities for two events, series of independent events, and normal distributions.

Probability of Two Independent Events

Calculate union, intersection, and other related probabilities of two independent events.

Results

P(A') 0
P(B') 0
P(A ∩ B) 0
P(A ∪ B) 0
P(A Δ B) (XOR) 0
P(neither A nor B) 0
P(A and not B) 0
P(B and not A) 0

Probability of a Series of Independent Events

Calculate binomial probabilities for repeated independent trials (e.g., probability of exactly k successes in n trials).

Result
0
Mean (np): 0
Variance (np(1-p)): 0

Probability of a Normal Distribution

Calculate the probability (area under the curve) between two bounds in a normal distribution, with visualization and confidence intervals.

Probability between bounds
0
Z-Left: 0 | Z-Right: 0

Common Confidence Intervals

Confidence Range
68.27%μ ± 1σ
95.45%μ ± 2σ
99.73%μ ± 3σ
95%μ ± 1.96σ
99%μ ± 2.58σ

Normal Distribution Curve

Probability Calculator Guide

1. Probability of Two Independent Events

This calculator computes all major probabilities related to two independent events A and B.

Key Formulas:

P(A') = 1 - P(A)
P(B') = 1 - P(B)
P(A ∩ B) = P(A) × P(B)
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A Δ B) = P(A) + P(B) - 2×P(A ∩ B)
P(neither) = 1 - P(A ∪ B)
P(A and not B) = P(A) × (1 - P(B))
P(B and not A) = (1 - P(A)) × P(B)

2. Series of Independent Events (Binomial Distribution)

This section calculates binomial probabilities — the probability of getting exactly k successes in n independent trials, each with success probability p. You can also calculate "at least k" or "at most k" successes.

P(X = k) = C(n,k) × p^k × (1-p)^(n-k)

Where C(n,k) = n! / (k!(n-k)!)

This is extremely useful for modeling coin flips, quality control, survey responses, and many other repeated independent trials.

3. Normal Distribution

Calculates the probability (area under the bell curve) between any two values. The normal distribution is the most important probability distribution in statistics.

Key Features of This Calculator:

  • Computes the exact probability between any Left and Right bound.
  • Shows the corresponding Z-scores (standardized values).
  • Visualizes the normal curve with the probability area highlighted.
  • Includes a reference table of common confidence intervals.

The 68-95-99.7 Rule:

  • ≈68.27% of data falls within 1 standard deviation of the mean.
  • ≈95.45% falls within 2 standard deviations.
  • ≈99.73% falls within 3 standard deviations.