Hypergeometrical Distribution

When data scientists are carrying out sampling without replacement, they will mostly use the hypergeometric distribution to determine the respective probabilities. The variables sampled are mainly referred to as hypergeometric variables.

Definition

Let;    N be the population size

          n be the sample size

          k be the number of classified successes in the population

          x be the number of classified successes in the sample

          kCx be the combination of k things taken x at a time

          h (x;N,n,k) be the hypergeometric probability

Given the above parameters, the hypergeometric probability can be expressed as follows;

h(x; N, n, k) = [ _kC_x ] [ _N-kC_n-x ] / [ _NC_n ]

From the above formula, the hypergeometric probability can simply be stated as the probability of obtaining x success without replacement from a sample size of n, given that the population size of N has k successes.

Properties

Mean = n*K/N

Variance = n * k * ( N - k ) * ( N - n ) / [ N² * ( N - 1 ) ] .

Example

Without replacement, a student randomly selects three cards from a standard deck of cards. Find the probability of drawing exactly two aces.

Answer

A standard deck has 52 cards. The number of aces in a standard deck is 4. Thus, our parameters will be as follows;

Applying the hypergeometric formula yields the following;

h(2;52,3,4)= 6*48/22100=0.01303

Conclusion

From the above result, we have that the probability of randomly drawing two aces from a standard deck of playing cards without replacement is 0.01303. This translates to a 1.303 % chance of obtaining that desired outcome.