Question

The Galton Board consists of a vertical board with interleaved rows of pegs. Beads are dropped from the top, and bounce either left or right as they hit the pegs. The bead bounces left or right with probability 1/2. Eventually the bead lands somewhere on the bottom.

Suppose that there are n rows of pegs. Define a sample space Ω and a probability law P which describes a single bead going through the Galton Board.

answer correctly or do not answer.

Answer 1

Sample Space Ω for n= 5 is given above.

The probability of landing in k^th bin of a system with n rows, in which the probability of going to the right at each peg is p and the probability of going to the left is 1-p. We multiply the number of paths, , times the probability of going right, p, to the k^th power, times the probability of going left, (1-p) to the (n-k)^th power (because if you go right k times, you necessarily go left the rest of the time). The probability of landing in the k^th bin is then:

× p^k × (1-p)^(n-k)​

Here p= 0.5 , 1-p = 0.5

So P[k] =