Statistics Examples

$x > 0$ , $n = 3$ , $p = 0.9$

Step 1

Subtract $0.9$ from $1$ .

$0.1$

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, $p (x > 0) = P (x = 1) + P (x = 2) + P (x = 3)$ .

$p (x > 0) = P (x = 1) + P (x = 2) + P (x = 3)$

Step 3

Find the probability of $P (1)$ .

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Step 3.1

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

$p (x) = {}_{3}C_{1} \cdot p^{x} \cdot q^{n - x}$

Step 3.2

Find the value of ${}_{3}C_{1}$ .

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Step 3.2.1

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

${}_{3}C_{1} = {}_{n}C_{r} = \frac{n!}{(r)! (n - r)!}$

Step 3.2.2

Fill in the known values.

$\frac{(3)!}{(1)! (3 - 1)!}$

Step 3.2.3

Simplify.

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Step 3.2.3.1

Subtract $1$ from $3$ .

$\frac{(3)!}{(1)! (2)!}$

Step 3.2.3.2

Rewrite $(3)!$ as $3 \cdot 2!$ .

$\frac{3 \cdot 2!}{(1)! (2)!}$

Step 3.2.3.3

Cancel the common factor of $2!$ .

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Step 3.2.3.3.1

Cancel the common factor.

$\frac{3 \cdot 2!}{(1)! (2)!}$

Step 3.2.3.3.2

Rewrite the expression.

$\frac{3}{(1)!}$

Step 3.2.3.4

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

$\frac{3}{1}$

Step 3.2.3.5

Divide $3$ by $1$ .

$3$

Step 3.3

Fill the known values into the equation.

$3 \cdot (0.9) \cdot {(1 - 0.9)}^{3 - 1}$

Step 3.4

Simplify the result.

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Step 3.4.1

Evaluate the exponent.

$3 \cdot 0.9 \cdot {(1 - 0.9)}^{3 - 1}$

Step 3.4.2

Multiply $3$ by $0.9$ .

$2.7 \cdot {(1 - 0.9)}^{3 - 1}$

Step 3.4.3

Subtract $0.9$ from $1$ .

$2.7 \cdot {0.1}^{3 - 1}$

Step 3.4.4

Subtract $1$ from $3$ .

$2.7 \cdot {0.1}^{2}$

Step 3.4.5

Raise $0.1$ to the power of $2$ .

$2.7 \cdot 0.01$

Step 3.4.6

Multiply $2.7$ by $0.01$ .

$0.027$

Step 4

Find the probability of $P (2)$ .

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Step 4.1

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

$p (x) = {}_{3}C_{2} \cdot p^{x} \cdot q^{n - x}$

Step 4.2

Find the value of ${}_{3}C_{2}$ .

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Step 4.2.1

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

${}_{3}C_{2} = {}_{n}C_{r} = \frac{n!}{(r)! (n - r)!}$

Step 4.2.2

Fill in the known values.

$\frac{(3)!}{(2)! (3 - 2)!}$

Step 4.2.3

Simplify.

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Step 4.2.3.1

Subtract $2$ from $3$ .

$\frac{(3)!}{(2)! (1)!}$

Step 4.2.3.2

Rewrite $(3)!$ as $3 \cdot 2!$ .

$\frac{3 \cdot 2!}{(2)! (1)!}$

Step 4.2.3.3

Cancel the common factor of $2!$ .

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Step 4.2.3.3.1

Cancel the common factor.

$\frac{3 \cdot 2!}{(2)! (1)!}$

Step 4.2.3.3.2

Rewrite the expression.

$\frac{3}{(1)!}$

Step 4.2.3.4

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

$\frac{3}{1}$

Step 4.2.3.5

Divide $3$ by $1$ .

$3$

Step 4.3

Fill the known values into the equation.

$3 \cdot {(0.9)}^{2} \cdot {(1 - 0.9)}^{3 - 2}$

Step 4.4

Simplify the result.

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Step 4.4.1

Raise $0.9$ to the power of $2$ .

$3 \cdot 0.81 \cdot {(1 - 0.9)}^{3 - 2}$

Step 4.4.2

Multiply $3$ by $0.81$ .

$2.43 \cdot {(1 - 0.9)}^{3 - 2}$

Step 4.4.3

Subtract $0.9$ from $1$ .

$2.43 \cdot {0.1}^{3 - 2}$

Step 4.4.4

Subtract $2$ from $3$ .

$2.43 \cdot {0.1}^{1}$

Step 4.4.5

Evaluate the exponent.

$2.43 \cdot 0.1$

Step 4.4.6

Multiply $2.43$ by $0.1$ .

$0.243$

Step 5

Find the probability of $P (3)$ .

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Step 5.1

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

$p (x) = {}_{3}C_{3} \cdot p^{x} \cdot q^{n - x}$

Step 5.2

Find the value of ${}_{3}C_{3}$ .

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Step 5.2.1

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

${}_{3}C_{3} = {}_{n}C_{r} = \frac{n!}{(r)! (n - r)!}$

Step 5.2.2

Fill in the known values.

$\frac{(3)!}{(3)! (3 - 3)!}$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Cancel the common factor of $(3)!$ .

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Step 5.2.3.1.1

Cancel the common factor.

$\frac{(3)!}{(3)! (3 - 3)!}$

Step 5.2.3.1.2

Rewrite the expression.

$\frac{1}{(3 - 3)!}$

Step 5.2.3.2

Simplify the denominator.

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Step 5.2.3.2.1

Subtract $3$ from $3$ .

$\frac{1}{(0)!}$

Step 5.2.3.2.2

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

$\frac{1}{1}$

Step 5.2.3.3

Divide $1$ by $1$ .

$1$

Step 5.3

Fill the known values into the equation.

$1 \cdot {(0.9)}^{3} \cdot {(1 - 0.9)}^{3 - 3}$

Step 5.4

Simplify the result.

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Step 5.4.1

Multiply ${(0.9)}^{3}$ by $1$ .

${(0.9)}^{3} \cdot {(1 - 0.9)}^{3 - 3}$

Step 5.4.2

Raise $0.9$ to the power of $3$ .

$0.729 \cdot {(1 - 0.9)}^{3 - 3}$

Step 5.4.3

Subtract $0.9$ from $1$ .

$0.729 \cdot {0.1}^{3 - 3}$

Step 5.4.4

Subtract $3$ from $3$ .

$0.729 \cdot {0.1}^{0}$

Step 5.4.5

Anything raised to $0$ is $1$ .

$0.729 \cdot 1$

Step 5.4.6

Multiply $0.729$ by $1$ .

$0.729$

Step 6

The probability $P (x > 0)$ is the sum of the probabilities of all possible $x$ values between $0$ and $n$ . $P (x > 0) = P (x = 1) + P (x = 2) + P (x = 3) = 0.027 + 0.243 + 0.729 = 0.999$ .

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Step 6.1

Add $0.027$ and $0.243$ .

$p (x > 0) = 0.27 + 0.729$

Step 6.2

Add $0.27$ and $0.729$ .

$p (x > 0) = 0.999$

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