Question

From the set {1,2,3,…,n},k distinct integers are selected at random and arranged in numerical order(from lowest to highest). Let P(i,r,k,n) denote the probability that integer i is in position r. For example, observe that P(1,2,k,n)=0, as it is possible for the number 1 to be in the second position after ordering.

(a) Compute P(2,1,6,10).
(b) Find a general formula for P(i,r,k,n).

Answer 1

To compute P(i, r, k, n), use the formula P(i, r, k, n) = 1/nCk. For P(2, 1, 6, 10), the probability is 0.00476. The general formula is P(i, r, k, n) = 1/((n!)/(k!(n-k)!)).

Step-by-step explanation:

To compute the probability that an integer i is in position r, we use the formula P(i, r, k, n) = 1/nCk. Where nCk is the binomial coefficient and represents the number of ways to choose k items from a set of n. In this case, P(2, 1, 6, 10) = 1/10C6 = 1/(10!/6!(10-6)!) = 1/210 = 0.00476.

For a general formula, P(i, r, k, n) = 1/nCk = 1/((n!)/(k!(n-k)!)). This formula represents the probability of selecting the integer i in the position r when randomly selecting k distinct integers from the set {1, 2, 3, ..., n} and arranging them in numerical order.