A bowl contains n lottery tickets numbered 1, 2,…, n.We select a ticket at random, record the number on it, and put it back in the bowl. The same procedure is followed a total of k times.

Find the probability that

(i) the number 1 is selected at least once;

(ii) both numbers 1 and 2 are selected at least once.

(Hint: Define the events Ai ∶ number i does not appear in any of the k selections for i = 1, 2. Then, the required probabilities are P(A′

1) and P(A′

1A′

2) = 1 − P(A1 ∪ A2).)

14. Let X = {x1, x2,…, xn} be a finite set with n ≥ 2 (distinct) elements. Suppose that we want to use

• r1 times the element x1,

• r2 times the element x2,

…

• rn times the element xn in order to form a collection (a1, a2,…, ar) of r units, where r = r1 + r2 + · · · + rn.

Such an arrangement is called an r-permutation of n (distinct) elements. Show that the number of all possible permutations of this type is given by the formula

Application: A part of the human DNA chain is represented as a series with elements A,C,G, T (the letters stand for the nucleobases adenine, cytosine, guanine, and thymine, respectively). How many different compositions (sequences) are possible for a segment of length r, such that r1 elements are of type A, r2 elements are of type C, r3 elements are of type G, and r4 elements are of type T (r = r1 + r2 + r3 + r4)?
Under the assumption that all such compositions have the same probability of appearing, what is the probability that a randomly selected sequence has the elements corresponding to each of the four bases being adjacent, i.e. we have compositions as in the following examples

Transcribed Image Text:

(tm)-w r!