Binomial Distribution Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

medium
A fair coin is flipped 88 times. What is the probability of getting exactly 55 heads?

Solution

  1. 1
    Use the binomial formula: P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, where n=8n = 8, k=5k = 5, p=0.5p = 0.5.
  2. 2
    (85)=8!5!⋅3!=56\binom{8}{5} = \frac{8!}{5! \cdot 3!} = 56.
  3. 3
    P(X=5)=56×(0.5)5×(0.5)3=56×(0.5)8=56×1256=56256=732P(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)=732≈0.219P(X = 5) = \frac{7}{32} \approx 0.219
The binomial distribution models the number of successes in nn independent trials, each with the same probability pp of success. The combination (nk)\binom{n}{k} counts the number of ways to arrange kk successes among nn trials.

About Binomial Distribution

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

Learn more about Binomial Distribution →

More Binomial Distribution Examples

Example 2 hard

A multiple-choice quiz has [formula] questions, each with [formula] choices. If a student guesses ra

Example 3 medium

A basketball player has a [formula] free-throw success rate. In [formula] attempts, what is the prob

Example 4 medium

A student guesses on [formula] true-or-false questions. What is the probability of getting at least

← All Binomial Distribution Examples Binomial Distribution Concept