Binomial Coefficient Math Example 2

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Example 2

hard
A fair coin is flipped 5 times. Using P(X=k)=(nk)pk(1−p)n−kP(X=k) = \binom{n}{k}p^k(1-p)^{n-k}, find P(X=3)P(X=3) (exactly 3 heads).

Solution

  1. 1
    Parameters: n=5n=5, k=3k=3, p=0.5p=0.5
  2. 2
    Calculate (53)=5!3!2!=1206×2=10\binom{5}{3} = \frac{5!}{3!2!} = \frac{120}{6 \times 2} = 10
  3. 3
    Calculate pk(1−p)n−k=(0.5)3(0.5)2=(0.5)5=132p^k(1-p)^{n-k} = (0.5)^3(0.5)^2 = (0.5)^5 = \frac{1}{32}
  4. 4
    P(X=3)=10×132=1032=0.3125P(X=3) = 10 \times \frac{1}{32} = \frac{10}{32} = 0.3125

Answer

P(X=3)=(53)(0.5)5=1032=0.3125P(X=3) = \binom{5}{3}(0.5)^5 = \frac{10}{32} = 0.3125
The binomial probability formula combines two parts: (nk)\binom{n}{k} counts the number of ways to arrange kk successes, and pk(1−p)n−kp^k(1-p)^{n-k} gives the probability of one specific arrangement. Multiplying gives total probability for exactly kk successes.

About Binomial Coefficient

The binomial coefficient (nk)\binom{n}{k} counts the number of ways to choose kk items from nn distinct items without regard to order. It equals n!k!(n−k)!\frac{n!}{k!(n-k)!}.

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More Binomial Coefficient Examples

Example 1 medium

Calculate [formula] using the formula [formula], and verify by listing all combinations of 2 items f

Example 3 medium

Calculate [formula] and explain what it counts.

Example 4 easy

Calculate: (a) [formula], (b) [formula], (c) [formula].

Example 5 hard

A committee of 3 is chosen from 8 people. How many possible committees exist? If one specific pair (

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