Statistics Examples

Find the Probability P(x<3) of the Binomial Distribution
, ,
Step 1
Subtract from .
Step 2
When the value of the number of successes is given as an interval, then the probability of is the sum of the probabilities of all possible values between and . In this case, .
Step 3
Find the probability of .
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Step 3.1
Use the formula for the probability of a binomial distribution to solve the problem.
Step 3.2
Find the value of .
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Step 3.2.1
Find the number of possible unordered combinations when items are selected from available items.
Step 3.2.2
Fill in the known values.
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 to .
Step 3.2.3.1.2
Multiply .
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Step 3.2.3.1.2.1
Multiply by .
Step 3.2.3.1.2.2
Multiply by .
Step 3.2.3.2
Simplify the denominator.
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Step 3.2.3.2.1
Expand to .
Step 3.2.3.2.2
Subtract from .
Step 3.2.3.2.3
Expand to .
Step 3.2.3.2.4
Multiply .
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Step 3.2.3.2.4.1
Multiply by .
Step 3.2.3.2.4.2
Multiply by .
Step 3.2.3.2.5
Multiply by .
Step 3.2.3.3
Divide by .
Step 3.3
Fill the known values into the equation.
Step 3.4
Simplify the result.
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Step 3.4.1
Multiply by .
Step 3.4.2
Anything raised to is .
Step 3.4.3
Multiply by .
Step 3.4.4
Subtract from .
Step 3.4.5
Subtract from .
Step 3.4.6
Raise to the power of .
Step 4
Find the probability of .
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Step 4.1
Use the formula for the probability of a binomial distribution to solve the problem.
Step 4.2
Find the value of .
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Step 4.2.1
Find the number of possible unordered combinations when items are selected from available items.
Step 4.2.2
Fill in the known values.
Step 4.2.3
Simplify.
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Step 4.2.3.1
Subtract from .
Step 4.2.3.2
Rewrite as .
Step 4.2.3.3
Cancel the common factor of .
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Step 4.2.3.3.1
Cancel the common factor.
Step 4.2.3.3.2
Rewrite the expression.
Step 4.2.3.4
Expand to .
Step 4.2.3.5
Divide by .
Step 4.3
Fill the known values into the equation.
Step 4.4
Simplify the result.
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Step 4.4.1
Evaluate the exponent.
Step 4.4.2
Multiply by .
Step 4.4.3
Subtract from .
Step 4.4.4
Subtract from .
Step 4.4.5
Raise to the power of .
Step 4.4.6
Multiply by .
Step 5
Find the probability of .
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Step 5.1
Use the formula for the probability of a binomial distribution to solve the problem.
Step 5.2
Find the value of .
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Step 5.2.1
Find the number of possible unordered combinations when items are selected from available items.
Step 5.2.2
Fill in the known values.
Step 5.2.3
Simplify.
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Step 5.2.3.1
Subtract from .
Step 5.2.3.2
Rewrite as .
Step 5.2.3.3
Cancel the common factor of .
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Step 5.2.3.3.1
Cancel the common factor.
Step 5.2.3.3.2
Rewrite the expression.
Step 5.2.3.4
Expand to .
Step 5.2.3.5
Divide by .
Step 5.3
Fill the known values into the equation.
Step 5.4
Simplify the result.
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Step 5.4.1
Raise to the power of .
Step 5.4.2
Multiply by .
Step 5.4.3
Subtract from .
Step 5.4.4
Subtract from .
Step 5.4.5
Evaluate the exponent.
Step 5.4.6
Multiply by .
Step 6
The probability is the sum of the probabilities of all possible values between and . .
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Step 6.1
Add and .
Step 6.2
Add and .
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