Question

A soft drink manufacturer puts one of three different quotations on the inside of each bottle cap. Assuming that each of the quotations is equally likely to appear, what is the probability of receiving all three after purchasing nine of the soft drinks? What is the expected number of soft drinks that you have to purchase to have all three quotations? Start by identifying the absorbing stochastic matrix A. 0 A- 2 0 (Type integers or simplified fractions.) The probability of receiving all three after purchasing eight of the soft drinks is (Type an integer or a simplified fraction.) The expected number of soft drinks that you have to purchase to have all three quotation:s (Type an integer, proper fraction, or mixed number.) soft drink(s).

Answer 1

Given that

we have to A soft drink manufacturer puts one of three different quotations on the inside of each bottle cap. Assuming that each of the

Let E be the event that a person receive all three quotations after purchasing five of the soft drink. We see

The sample space is of size

n(E) -P(3,3)C (5,3)+ P(3,1)C(5,1)C(4,2) C(2,2 6 x 10+3x5x6 150

N-3 - 243

Recall the fundamental result for experiments with equally likely outcomes:" Let S be a sample space consisting of N equally likely outcomes. Let E be any event of size n(E). Then

" we see

Pr(E) N 150 81

We consider "3 quotations" as "A", "1 quotation" as "B" and "2 quotations" as "C". Based on the given information , we obtain the stochastic matrix for this Markov process is

, Note that the standard form of the absorbing stochastic matrix is

C130 23 B 01323 A 1-00

We see

is the standard form of the absorbing stochastic matrix.

C130 23 B 01323 A 1-00

By a graphing calculator, we see

Recall the interpretation of the entries of A". "The entry in the i^th row

5-1 0 6-8 13 0 23 1-31 0-7 0 1323 5818 10 27 0 100 100

and j^th column of the matrix A" is the probability of the transition from state. j to state i after n time periods. "We see the probability is of receiving all three after purchasing 4 + 1 = 5 of the soft drinks, which asserts our previous result.

58

If the stochastic matrix is partitioned in to submatrices

then the stable matrix of A is

. We see in this case

and

2 1 13 23

It follows that

Therefore we obtain the stable matrix of A is

0 23 13 23 1 3 23 23 3-2 3 13 ) 3 3 0

A B C C 0 0 0

We see the fundamental matrix for this Markov process is

0 23 13 23 1 23-32 3 ( (o 2-3 2-3

Recall the properties of the fundamental matrix  

F=(1-R)" -1

"The i^thentry of F is the expected number of times the process will be in nonabsorbing state i . if it starts in nonabsorbing state j.

The sum of the entries of the j^th column of F is the expected number of steps before 'ABSORPTION' When the process begins in nonabsorbing state j". We see the expected number of soft drinks you have to purchase to have all three quotations is the sum of entries in the first column of F adding 1. That is , the expected number is

n=-+3+1