Binomial Distribution Worksheet - Free Printable

1.
a) $\frac{5!}{3!2!} = \frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(3 \cdot 2 \cdot 1)(2 \cdot 1)} = \frac{120}{6 \cdot 2} = \frac{120}{12} = 10$
b) $\frac{7!}{7!0!} = \frac{5040}{5040 \cdot 1} = 1$

2.
a) $\binom{5}{0} = \frac{5!}{0!(5-0)!} = \frac{120}{1 \cdot 120} = 1$
b) $\binom{5}{1} = \frac{5!}{1!(5-1)!} = \frac{120}{1 \cdot 24} = 5$
c) $\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{120}{2 \cdot 6} = \frac{120}{12} = 10$
d) $\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{120}{6 \cdot 2} = \frac{120}{12} = 10$
e) $\binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{120}{24 \cdot 1} = 5$
f) $\binom{5}{5} = \frac{5!}{5!(5-5)!} = \frac{120}{120 \cdot 1} = 1$

3.
$\sum_{x=0}^{5} \binom{5}{x} = \binom{5}{0} + \binom{5}{1} + \binom{5}{2} + \binom{5}{3} + \binom{5}{4} + \binom{5}{5} = 1 + 5 + 10 + 10 + 5 + 1 = 32 = 2^5$

4.
The binomial probability formula is $P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$, where $n = 4$, $p = 0.12$, and $1-p = 0.88$.

- $P(X = 0) = \binom{4}{0} (0.12)^0 (0.88)^4 = 1 \cdot 1 \cdot 0.5997 = 0.5997$
- $P(X = 1) = \binom{4}{1} (0.12)^1 (0.88)^3 = 4 \cdot 0.12 \cdot 0.6815 = 4 \cdot 0.08178 = 0.3271$
- $P(X = 2) = \binom{4}{2} (0.12)^2 (0.88)^2 = 6 \cdot 0.0144 \cdot 0.7744 = 6 \cdot 0.01115 = 0.0669$
- $P(X = 3) = \binom{4}{3} (0.12)^3 (0.88)^1 = 4 \cdot 0.001728 \cdot 0.88 = 4 \cdot 0.00152 = 0.0061$
- $P(X = 4) = \binom{4}{4} (0.12)^4 (0.88)^0 = 1 \cdot 0.000207 \cdot 1 = 0.000207$

| $x$ | $P(X = x)$ |
|-----|------------|
| 0 | 0.5997 |
| 1 | 0.3271 |
| 2 | 0.0669 |
| 3 | 0.0061 |
| 4 | 0.0002 |