Finite Math Examples

$n = 14$ , $x = 5$ , $p = 0.2$

Step 1

Use the formula for the probability of a binomial distribution to solve the problem.

$p (x) = {}_{14}C_{5} \cdot p^{x} \cdot q^{n - x}$

Step 2

Find the value of ${}_{14}C_{5}$ .

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Step 2.1

Find the number of possible unordered combinations when $r$ items are selected from $n$ available items.

${}_{14}C_{5} = {}_{n}C_{r} = \frac{n!}{(r)! (n - r)!}$

Step 2.2

Fill in the known values.

$\frac{(14)!}{(5)! (14 - 5)!}$

Step 2.3

Simplify.

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Step 2.3.1

Subtract $5$ from $14$ .

$\frac{(14)!}{(5)! (9)!}$

Step 2.3.2

Rewrite $(14)!$ as $14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9!$ .

$\frac{14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9!}{(5)! (9)!}$

Step 2.3.3

Cancel the common factor of $9!$ .

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Step 2.3.3.1

Cancel the common factor.

$\frac{14 \cdot 13 \cdot 12 \cdot 11 \cdot 10 \cdot 9!}{(5)! (9)!}$

Step 2.3.3.2

Rewrite the expression.

$\frac{14 \cdot 13 \cdot 12 \cdot 11 \cdot 10}{(5)!}$

Step 2.3.4

Simplify the numerator.

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Step 2.3.4.1

Multiply $14$ by $13$ .

$\frac{182 \cdot 12 \cdot 11 \cdot 10}{(5)!}$

Step 2.3.4.2

Multiply $182$ by $12$ .

$\frac{2184 \cdot 11 \cdot 10}{(5)!}$

Step 2.3.4.3

Multiply $2184$ by $11$ .

$\frac{24024 \cdot 10}{(5)!}$

Step 2.3.4.4

Multiply $24024$ by $10$ .

$\frac{240240}{(5)!}$

Step 2.3.5

Simplify the denominator.

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Step 2.3.5.1

Expand $(5)!$ to $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

$\frac{240240}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}$

Step 2.3.5.2

Multiply $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

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Step 2.3.5.2.1

Multiply $5$ by $4$ .

$\frac{240240}{20 \cdot 3 \cdot 2 \cdot 1}$

Step 2.3.5.2.2

Multiply $20$ by $3$ .

$\frac{240240}{60 \cdot 2 \cdot 1}$

Step 2.3.5.2.3

Multiply $60$ by $2$ .

$\frac{240240}{120 \cdot 1}$

Step 2.3.5.2.4

Multiply $120$ by $1$ .

$\frac{240240}{120}$

Step 2.3.6

Divide $240240$ by $120$ .

$2002$

Step 3

Fill the known values into the equation.

$2002 \cdot {(0.2)}^{5} \cdot {(1 - 0.2)}^{14 - 5}$

Step 4

Simplify the result.

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Step 4.1

Raise $0.2$ to the power of $5$ .

$2002 \cdot 0.00032 \cdot {(1 - 0.2)}^{14 - 5}$

Step 4.2

Multiply $2002$ by $0.00032$ .

$0.64064 \cdot {(1 - 0.2)}^{14 - 5}$

Step 4.3

Subtract $0.2$ from $1$ .

$0.64064 \cdot {0.8}^{14 - 5}$

Step 4.4

Subtract $5$ from $14$ .

$0.64064 \cdot {0.8}^{9}$

Step 4.5

Raise $0.8$ to the power of $9$ .

$0.64064 \cdot 0.13421772$

Step 4.6

Multiply $0.64064$ by $0.13421772$ .

$0.08598524$