Reference no: EM131833123

In his book Natural Inheritance, p. 63, F. Galton in 1889 described an apparatus known today as Galton's quincunx. The apparatus consists of a board in which nails are arranged in rows, the nails of a given row being placed below the mid-points of the intervals between the nails in the row above. Small steel balls of equal diameter are poured into the apparatus through a funnel located opposite the central pin of the first row. As they run down the board, the balls are "influenced" by the nails in such a manner that, after passing through the last row, they take up positions deviating from the point vertically below the central pin of the first row. Let us call this point .1: = O. Assume that the distance between 2 neighboring pins is taken to be I and that the diameter of the balls is slightly smaller than 1. Assume that in passing from one row, to the next the abscissa (x-coordinate) of a ball changes by either ½ or -½, each possibility having equal probability. To each opening in a row of nails, assign as its abscissa the mid-point of the interval between the 2 nails. If there is an even number of rows of nails, then the openings in the last row will have abscissas 0, ± I, ±2, . . .. Assuming that there are 36 rows of nails, find for k = 0, ± I, ±2, ... , ± 10 the probability that a ball inserted in the funnel will pass through the opening in the last row, which has abscissa k.