STEM Calculators

Probability of Success and the Shape of Binomial Distribution

Statistics • Discrete Random Variables and Their Probability Distributions

Written by STEM Calculators Team Published Updated
View all topics

Probability of Success and the Shape of the Binomial Distribution

Enter the number of trials n and the probability of success p. The tool builds the full binomial probability distribution P(x) for x = 0, 1, 2, …, n and draws a bar graph. It also explains the shape based on p.

Shape rules For any fixed n:
  • The distribution is symmetric if p = .50.
  • It is skewed to the right if p < .50.
  • It is skewed to the left if p > .50.
Typical values: 1–60. Larger n works, but bars become very thin.
Failure probability: q = 1 − p → 0.50
Drag to see the distribution change smoothly.
If blank, the tool highlights a reasonable x automatically.
Ready
Enter n and p, then press Calculate.

Rate this calculator

0.0 /5 (0 ratings)
Be the first to rate.
Your rating
You can update your rating any time.

Frequently Asked Questions

How does p affect the shape of a binomial distribution?

For a fixed n, the distribution is symmetric when p = 0.50. It is right-skewed when p < 0.50 and left-skewed when p > 0.50.

What is q in a binomial distribution?

q is the probability of failure on each trial and equals 1 - p. The binomial formula uses both p and q to compute P(x).

What formula is used to compute P(x) in a binomial distribution?

The binomial probability is P(x) = C(n,x) x p^x x q^(n - x), where n is the number of trials, x is the number of successes, p is the success probability, and q = 1 - p.

Why does the binomial distribution become skewed when p is not 0.50?

When p moves away from 0.50, the probability mass shifts toward smaller x values (if p < 0.50) or toward larger x values (if p > 0.50). This creates a longer tail on one side of the bar graph.

Related calculators