Science:Math Exam Resources/Courses/MATH152/April 2022/Question B2 (b)

MATH152 April 2022

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Question B2 (b)
This is a continuation of B2 (a). (b) Find the probability that a Braeburn buyer will buy Braeburn again after two weeks.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
The probability of that a Braeburn buyer will buy a Braeburn again in two weeks is the probability of starting at vertex B and ending up at vertex B again after two steps. How many walks are there that take two steps, start at B and end at B?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. There are 3 ways to walk from B to B in two steps: $$B\rightarrow B\rightarrow B,\qquad B\rightarrow A\rightarrow B,\qquad B\rightarrow C\rightarrow B.$$ Each individual path has probability $p_{2,i}\cdot p_{i,2}$, for $i=1,2,3$, and where $p_{i,j}$ is the probability of going from vertex $j$ to vertex $i$ (which is also the $(i,j)$ entry of the transition matrix computed in the previous question). The total probability of starting at B and ending at B after two steps is the sum $$p_{2,1}\cdot p_{1,2}+p_{2,2}\cdot p_{2,2}+p_{2,3}\cdot p_{3,2}=0.2\cdot 0.2+0.6\cdot 0.6+0.4\cdot 0.2=0.48,$$ which is the (2,2) entry of the transition matrix squared.