Hypergeometric Distribution - Order of Draws

Order of Draws

The probability of drawing any sequence of white and black marbles (the hypergeometric distribution) depends only on the number of white and black marbles, not on the order in which they appear; i.e., it is an exchangeable distribution. As a result, the probability of drawing a white marble in the draw is

This can be shown by induction. First, it is certainly true for the first draw that:

.

Also, we can show that by writing:

\begin{align} P(W_{n+1}) & = {\sum_{k=0}^n}P(W_{n+1}|k)f(k;N,m,n)\\ & = {\sum_{k=0}^n}\frac{m-k}{N-n}f(k;N,m,n) \\ & = {\sum_{k=0}^n}\frac{m-k}{N-n}\frac{\binom mk \binom {N-m} {n-k}}{\binom Nn} \\ & = \frac{1}{(N-n)\binom Nn} \left \{ m\sum_{k=0}^n \binom mk \binom {N-m} {n-k} - \sum_{k=0}^n k\binom mk \binom {N-m} {n-k}\right \} \\ & = \frac{1}{(N-n)\binom Nn}\left\{ m\binom Nn - \sum_{k=1}^n k\frac{m}{k} \binom {m-1}{k-1} \binom {N-m} {n-k}\right \} \\ & = \frac{m}{(N-n)\binom Nn}\left\{ \binom Nn - \sum_{k=1}^n \binom {m-1}{k-1} \binom {N-1-(m-1)} {n-1-(k-1)}\right \} \\ & = \frac{m}{(N-n)\binom Nn}\left\{ \binom Nn - \binom {N-1}{n-1}\right \} \\ & = \frac{m}{(N-n)\binom Nn}\left\{ \binom Nn - \frac{n}{N}\binom Nn\right \} \\ & = \frac{m}{(N-n)}\left\{ 1 - \frac{n}{N} \right\} = \frac{m}{N} \end{align} ,

which makes it true for every draw.

A simpler proof than the one above is the following:

By symmetry each of the marbles has the same chance to be drawn in the draw. In addition, according to the sum rule, the chance of drawing a white marble in the draw can be calculated by summing the chances of each individual white marble being drawn in the . These two observations imply that if for example the number of white marbles at the outset is 3 times the number of black marbles, then also the chance of a white marble being drawn in the draw is 3 times as big as a black marble being drawn in the draw. In the general case we have white marbles and black marbles at the outset. So

.

Since in the draw either a white or a black marble needs to be drawn, we also know that

.

Combining these two equations immediately yields

.

Read more about this topic:  Hypergeometric Distribution

Famous quotes containing the words order of, order and/or draws:

    “There is no permanent class of hired laborers amongst us. Twenty five years ago, I was a hired laborer. The hired laborer of yesterday, labors on his own account today; and will hire others to labor for him tomorrow. Advancement—improvement in condition—is the order of things in a society of equals.”
    —Abraham Lincoln (1809–1865)

    “We have created an industrial order geared to automatism, where feeble-mindedness, native or acquired, is necessary for docile productivity in the factory; and where a pervasive neurosis is the final gift of the meaningless life that issues forth at the other end.”
    —Lewis Mumford (1895–1990)

    “Our day of dependence, our long apprenticeship to the learning of other lands, draws to a close. The millions, that around us are rushing into life, cannot always be fed on the sere remains of foreign harvests.”
    —Ralph Waldo Emerson (1803–1882)