A hypergeometric experiment is conducted with the given parameters. Compute the probability of the random variable X.
[tex]$N=120, n=23, k=25, X=7$[/tex]
[tex]$P ( x ) \approx$[/tex] (Round to four decimal places as needed.)
Answer (2)
Use the hypergeometric probability formula: P ( X = x ) = ( n N ) ( x k ) ( n − x N − k ) .
Substitute the given values: P ( X = 7 ) = ( 23 120 ) ( 7 25 ) ( 23 − 7 120 − 25 ) = ( 23 120 ) ( 7 25 ) ( 16 95 ) .
Calculate the binomial coefficients and compute the probability: P ( X = 7 ) ≈ 0.0986186778 .
Round to four decimal places: P ( X = 7 ) ≈ 0.0986 .
Explanation
Understand the problem and provided data We are given a hypergeometric experiment with the following parameters:
N = 120 (population size) n = 23 (sample size) k = 25 (number of success states in the population) X = 7 (number of success states in the sample)
We want to compute the probability P ( X = 7 ) .
State the formula The hypergeometric probability formula is given by:
P ( X = x ) = ( n N ) ( x k ) ( n − x N − k )
where:
N is the population size,
n is the sample size,
k is the number of success states in the population,
x is the number of success states in the sample.
Plug in the values Plugging in the given values, we have:
P ( X = 7 ) = ( 23 120 ) ( 7 25 ) ( 23 − 7 120 − 25 ) = ( 23 120 ) ( 7 25 ) ( 16 95 )
Now we calculate the binomial coefficients.
Calculate the probability Using a calculator, we find:
( 7 25 ) = 2422780 ( 16 95 ) = 122747184775 ( 23 120 ) = 2963735489960
Therefore,
P ( X = 7 ) = 2963735489960 2422780 × 122747184775 = 2963735489960 297387984974550 ≈ 0.0986186778
Round the result Rounding the result to four decimal places, we get:
P ( X = 7 ) ≈ 0.0986
Final Answer The probability of the random variable X is approximately 0.0986.
Examples
Consider a quality control scenario where a batch of 120 items contains 25 defective items. If you randomly select 23 items, the probability of finding exactly 7 defective items in your sample can be calculated using the hypergeometric distribution. This helps assess the effectiveness of the sampling process and estimate the overall quality of the batch. Hypergeometric distribution is useful in scenarios where sampling is done without replacement, and the population size is finite.
The probability of drawing exactly 7 successes in a hypergeometric experiment with the given parameters is approximately 0.1154. This result is calculated using the hypergeometric probability formula. The answer is rounded to four decimal places.
;
