Finite Math Examples

$x < 1$ , $n = 12$ , $p = 0.7$

Step 1

Subtract $0.7$ from $1$ .

$0.3$

Step 2

When the value of the number of successes $x$ is given as an interval, then the probability of $x$ is the sum of the probabilities of all possible $x$ values between $0$ and $n$ . In this case, $p (x < 1) = P (x = 0)$ .

$p (x < 1) = P (x = 0)$

Step 3

Find the probability of $p (0)$ .

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Step 3.1

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

$p (x) = {}_{12}C_{0} \cdot p^{x} \cdot q^{n - x}$

Step 3.2

Find the value of ${}_{12}C_{0}$ .

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Step 3.2.1

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

${}_{12}C_{0} = {}_{n}C_{r} = \frac{n!}{(r)! (n - r)!}$

Step 3.2.2

Fill in the known values.

$\frac{(12)!}{(0)! (12 - 0)!}$

Step 3.2.3

Simplify.

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Step 3.2.3.1

Simplify the numerator.

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Step 3.2.3.1.1

Expand $(12)!$ to $12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

$\frac{12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(0)! (12 - 0)!}$

Step 3.2.3.1.2

Multiply $12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

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Step 3.2.3.1.2.1

Multiply $12$ by $11$ .

$\frac{132 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(0)! (12 - 0)!}$

Step 3.2.3.1.2.2

Multiply $132$ by $10$ .

$\frac{1320 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(0)! (12 - 0)!}$

Step 3.2.3.1.2.3

Multiply $1320$ by $9$ .

$\frac{11880 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(0)! (12 - 0)!}$

Step 3.2.3.1.2.4

Multiply $11880$ by $8$ .

$\frac{95040 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(0)! (12 - 0)!}$

Step 3.2.3.1.2.5

Multiply $95040$ by $7$ .

$\frac{665280 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(0)! (12 - 0)!}$

Step 3.2.3.1.2.6

Multiply $665280$ by $6$ .

$\frac{3991680 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(0)! (12 - 0)!}$

Step 3.2.3.1.2.7

Multiply $3991680$ by $5$ .

$\frac{19958400 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(0)! (12 - 0)!}$

Step 3.2.3.1.2.8

Multiply $19958400$ by $4$ .

$\frac{79833600 \cdot 3 \cdot 2 \cdot 1}{(0)! (12 - 0)!}$

Step 3.2.3.1.2.9

Multiply $79833600$ by $3$ .

$\frac{239500800 \cdot 2 \cdot 1}{(0)! (12 - 0)!}$

Step 3.2.3.1.2.10

Multiply $239500800$ by $2$ .

$\frac{479001600 \cdot 1}{(0)! (12 - 0)!}$

Step 3.2.3.1.2.11

Multiply $479001600$ by $1$ .

$\frac{479001600}{(0)! (12 - 0)!}$

Step 3.2.3.2

Simplify the denominator.

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Step 3.2.3.2.1

Expand $(0)!$ to $1$ .

$\frac{479001600}{1 (12 - 0)!}$

Step 3.2.3.2.2

Subtract $0$ from $12$ .

$\frac{479001600}{1 (12)!}$

Step 3.2.3.2.3

Expand $(12)!$ to $12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

$\frac{479001600}{1 (12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$

Step 3.2.3.2.4

Multiply $12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

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Step 3.2.3.2.4.1

Multiply $12$ by $11$ .

$\frac{479001600}{1 (132 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$

Step 3.2.3.2.4.2

Multiply $132$ by $10$ .

$\frac{479001600}{1 (1320 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$

Step 3.2.3.2.4.3

Multiply $1320$ by $9$ .

$\frac{479001600}{1 (11880 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$

Step 3.2.3.2.4.4

Multiply $11880$ by $8$ .

$\frac{479001600}{1 (95040 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$

Step 3.2.3.2.4.5

Multiply $95040$ by $7$ .

$\frac{479001600}{1 (665280 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$

Step 3.2.3.2.4.6

Multiply $665280$ by $6$ .

$\frac{479001600}{1 (3991680 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$

Step 3.2.3.2.4.7

Multiply $3991680$ by $5$ .

$\frac{479001600}{1 (19958400 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$

Step 3.2.3.2.4.8

Multiply $19958400$ by $4$ .

$\frac{479001600}{1 (79833600 \cdot 3 \cdot 2 \cdot 1)}$

Step 3.2.3.2.4.9

Multiply $79833600$ by $3$ .

$\frac{479001600}{1 (239500800 \cdot 2 \cdot 1)}$

Step 3.2.3.2.4.10

Multiply $239500800$ by $2$ .

$\frac{479001600}{1 (479001600 \cdot 1)}$

Step 3.2.3.2.4.11

Multiply $479001600$ by $1$ .

$\frac{479001600}{1 \cdot 479001600}$

Step 3.2.3.2.5

Multiply $479001600$ by $1$ .

$\frac{479001600}{479001600}$

Step 3.2.3.3

Divide $479001600$ by $479001600$ .

$1$

Step 3.3

Fill the known values into the equation.

$1 \cdot {(0.7)}^{0} \cdot {(1 - 0.7)}^{12 - 0}$

Step 3.4

Simplify the result.

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Step 3.4.1

Multiply ${(0.7)}^{0}$ by $1$ .

${(0.7)}^{0} \cdot {(1 - 0.7)}^{12 - 0}$

Step 3.4.2

Anything raised to $0$ is $1$ .

$1 \cdot {(1 - 0.7)}^{12 - 0}$

Step 3.4.3

Multiply ${(1 - 0.7)}^{12 - 0}$ by $1$ .

${(1 - 0.7)}^{12 - 0}$

Step 3.4.4

Subtract $0.7$ from $1$ .

${0.3}^{12 - 0}$

Step 3.4.5

Subtract $0$ from $12$ .

${0.3}^{12}$

Step 3.4.6

Raise $0.3$ to the power of $12$ .

$0.00000053$