Answered: Each time a shopper purchases a tube of…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Each time a shopper purchases a tube of toothpaste, he either choose brand A or brand B. Suppose that for each purchase after the 1st, the probability is 1/3 that he will choose the same brand that he chose on his preceding purchase and the probability is 2/3 that he will switch brands. If he's equally likely to choose either brand A or brand B on his first purchase, what is the probability that both his 1st and 2nd purchases will be brand A and both his 3rd and 4th purchases will be brand B?

Expert Solution

Step 1

Since the probability of purchase of any of the brands (i.e., "A" or "B") defines the probability of purchase of the next specific brand, hence to obtain the probability (using the transition probability matrix) of purchasing of specific brand at ${(r + 1)}^{t h}$ state, the given probability is given as:

$P_{r} [A B] \times [\begin{matrix} A & B \\ A & P_{A A} & P_{A B} \\ B & P_{B A} & P_{B B} \end{matrix}] w h e r e, P [A B] \begin{matrix} \end{matrix} = i n t i t i a l p r o b a b i l i t y o f p u r c h a \sin g e a c h b r a n d A a n d B P_{i j} (i = A, B; j = A, B) = p r o b a b i l i t y o f m o v i n g f r o m b r a n d i t o b r a n d j A n d P_{A A} + P_{A B} = P_{B A} + P_{B B} = 1$