Finite Math Examples

$x < 1$ , $n = 6$ , $p = 5$

Step 1

Subtract $5$ from $1$ .

$- 4$

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.

$p (x) = {}_{6}C_{0} \cdot p^{x} \cdot q^{n - x}$

Step 3.2

Find the value of ${}_{6}C_{0}$ .

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Step 3.2.1

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

${}_{6}C_{0} = {}_{n}C_{r} = \frac{n!}{(r)! (n - r)!}$

Step 3.2.2

Fill in the known values.

$\frac{(6)!}{(0)! (6 - 0)!}$

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 $(6)!$ to $6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

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

Step 3.2.3.1.2

Multiply $6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

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Step 3.2.3.1.2.1

Multiply $6$ by $5$ .

$\frac{30 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{(0)! (6 - 0)!}$

Step 3.2.3.1.2.2

Multiply $30$ by $4$ .

$\frac{120 \cdot 3 \cdot 2 \cdot 1}{(0)! (6 - 0)!}$

Step 3.2.3.1.2.3

Multiply $120$ by $3$ .

$\frac{360 \cdot 2 \cdot 1}{(0)! (6 - 0)!}$

Step 3.2.3.1.2.4

Multiply $360$ by $2$ .

$\frac{720 \cdot 1}{(0)! (6 - 0)!}$

Step 3.2.3.1.2.5

Multiply $720$ by $1$ .

$\frac{720}{(0)! (6 - 0)!}$

Step 3.2.3.2

Simplify the denominator.

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Step 3.2.3.2.1

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

$\frac{720}{1 (6 - 0)!}$

Step 3.2.3.2.2

Subtract $0$ from $6$ .

$\frac{720}{1 (6)!}$

Step 3.2.3.2.3

Expand $(6)!$ to $6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

$\frac{720}{1 (6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$

Step 3.2.3.2.4

Multiply $6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$ .

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Step 3.2.3.2.4.1

Multiply $6$ by $5$ .

$\frac{720}{1 (30 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$

Step 3.2.3.2.4.2

Multiply $30$ by $4$ .

$\frac{720}{1 (120 \cdot 3 \cdot 2 \cdot 1)}$

Step 3.2.3.2.4.3

Multiply $120$ by $3$ .

$\frac{720}{1 (360 \cdot 2 \cdot 1)}$

Step 3.2.3.2.4.4

Multiply $360$ by $2$ .

$\frac{720}{1 (720 \cdot 1)}$

Step 3.2.3.2.4.5

Multiply $720$ by $1$ .

$\frac{720}{1 \cdot 720}$

Step 3.2.3.2.5

Multiply $720$ by $1$ .

$\frac{720}{720}$

Step 3.2.3.3

Divide $720$ by $720$ .

$1$

Step 3.3

Fill the known values into the equation.

$1 \cdot {(5)}^{0} \cdot {(1 - 5)}^{6 - 0}$

Step 3.4

Simplify the result.

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Step 3.4.1

Multiply ${(5)}^{0}$ by $1$ .

${(5)}^{0} \cdot {(1 - 5)}^{6 - 0}$

Step 3.4.2

Anything raised to $0$ is $1$ .

$1 \cdot {(1 - 5)}^{6 - 0}$

Step 3.4.3

Multiply ${(1 - 5)}^{6 - 0}$ by $1$ .

${(1 - 5)}^{6 - 0}$

Step 3.4.4

Subtract $5$ from $1$ .

${(- 4)}^{6 - 0}$

Step 3.4.5

Subtract $0$ from $6$ .

${(- 4)}^{6}$

Step 3.4.6

Raise $- 4$ to the power of $6$ .

$4096$