There is a package containing 20 nuts and 20 bolts. A sample of eight is taken from the package at random. Let X be the number of nuts in the sample.
What is the name of the distribution for X?
I'm having trouble identifying distributions, can someone tell me the name of the distribution for X and why?
Please and thank you!
DrPhil said:
I don't know a NAME for this distribution - all I know is that it will involve Combinations, because we don't care what order the nuts and bolts are selected in.
How many ways can you select X nuts out of 20? How many ways can you select (8 - X) bolts out of 20? Since these two selections are independent of each other, the total is the product of the two numbers.
If you want probability, divide that product by the number of ways you can select 8 from a total of 40. Words that describe the distribution are symmetric, discrete, mean = mode = median = 4. By my calculation, P(4) = 0.305.
FOUND THE NAME OF THIS DISTRIBUTION!! It is a
HYPERGEOMETRIC DISTRIBUTION.
Having used "brute force" to come up with the product of two Combinations divided by a third Combination, I compared to formulas in the "Handbook of Probability and Statistics" to see if any matched. And found this:
\(\displaystyle \displaystyle f(x) = \dfrac{\binom k x \binom {N-k} {n-x}}{\binom N n} \),
...\(\displaystyle x = 0,1,2,. . ., min(n,k)\)
where N = total number of items in a finite population = 40,
.........n = number of items drawn without replacement = 8
.........k = number of successes (nuts) in finite population = 20
.........N - k = number of failures (not nuts) = 20
.........x = number of successes (nuts) in sample = 0, 1, 2, . . . , 8 (smaller of n or k)
.........n - x = number of failures (not nuts) in sample