Binomial Distribution Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Binomial Distribution.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The probability distribution of the number of successes in n independent yes/no trials, each with probability p.

Flip a biased coin n times—how many heads? The binomial distribution gives the probability of each count.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Each trial is independent with same probability. Count the successes, not the order.

Common stuck point: The \binom{n}{k} counts the arrangements—without it you'd only get one specific order's probability.

Worked Examples

Example 1

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A fair coin is flipped 8 times. What is the probability of getting exactly 5 heads?

Solution

  1. 1
    Use the binomial formula: P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, where n = 8, k = 5, p = 0.5.
  2. 2
    \binom{8}{5} = \frac{8!}{5! \cdot 3!} = 56.
  3. 3
    P(X = 5) = 56 \times (0.5)^5 \times (0.5)^3 = 56 \times (0.5)^8 = 56 \times \frac{1}{256} = \frac{56}{256} = \frac{7}{32}.

Answer

P(X = 5) = \frac{7}{32} \approx 0.219
The binomial distribution models the number of successes in n independent trials, each with the same probability p of success. The combination \binom{n}{k} counts the number of ways to arrange k successes among n trials.

Example 2

hard
A multiple-choice quiz has 10 questions, each with 4 choices. If a student guesses randomly, what is the probability of getting at least 2 correct?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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A basketball player has a 70\% free-throw success rate. In 6 attempts, what is the probability of making exactly 4?

Example 2

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A student guesses on 7 true-or-false questions. What is the probability of getting at least 6 correct?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

binomial coefficientprobabilityindependent events