Binomial Distribution Table

Binomial Distribution Table

The binomial table shows probabilities for the binomial distribution, which models the number of successes in a fixed number of independent trials.

Binomial Probability Formula

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

Where:

• n = number of trials
• k = number of successes
• p = probability of success on each trial

Cumulative Probabilities P(X ≤ k)

n = 5 trials

n = 10 trials

n = 15 trials

n = 20 trials

Individual Probabilities P(X = k)

n = 10, p = 0.50

How to Calculate Probabilities

Individual Probability: P(X = k)

Read directly from individual probability tables, or: $P(X = k) = P(X \leq k) - P(X \leq k-1)$

“At least” Probability: P(X ≥ k)

$P(X \geq k) = 1 - P(X \leq k-1)$

Range Probability: P(a ≤ X ≤ b)

$P(a \leq X \leq b) = P(X \leq b) - P(X \leq a-1)$

When to Use Binomial Distribution

✓ Fixed number of trials (n)
✓ Each trial has two outcomes (success/failure)
✓ Constant probability of success (p)
✓ Independent trials

Example Problems

Example 1: Exact Probability

Problem: What is P(X = 3) when n = 10, p = 0.30?

Solution:

• P(X ≤ 3) = 0.6496
• P(X ≤ 2) = 0.3828
• P(X = 3) = 0.6496 - 0.3828 = 0.2668

Example 2: At Least

Problem: What is P(X ≥ 4) when n = 10, p = 0.30?

Solution:

• P(X ≥ 4) = 1 - P(X ≤ 3)
• P(X ≥ 4) = 1 - 0.6496 = 0.3504

Binomial Mean and Standard Deviation

Related Resources

• Binomial Distribution Lesson - Complete guide
• Probability Calculator - Calculate probabilities
• Normal Approximation - When n is large
• Poisson Table - For rare events