Question

Problem 3. A video cassette recorder manufacturer is so certain of its quality control that it is offering a complete replacement warranty if a recorder fails within 2 years. Based upon compiled data, the company has noted that only 3 percent of its recorders fail during the first year, whereas 2 percent of the recorders that survive the first year will fail during the second year. The warranty does not cover replacement recorders. (a) Formulate the evolution of the status of a recorder as a Markov chain whose states include two absorption states that involve needing to honor the warranty or having the recorder survive the warranty period. Then construct the (one-step) transition matrix. (b) Find the probability that the manufacturer will have to honor the warranty.

Answer 1

(a)

The Markov chain for the given problem can be modeled with 4 states depicting the year after purchase and absorption states (honor the warranty and survive the warranty period). Let the states be S1, S2, Sw and Ss where S1 denotes the first year, S2 denotes the second year, Sw denotes the state that involve needing to honor the warranty, Ss denotes the recorder survive the warranty period.

The transition probability from State S1 to Sw is 0.03 (3% of recorders fail during first year). The transition probability of state S1 to state S2 is 1-0.03 = 0.97.

The transition probability from State S2 to Sw is 0.02 (2% of recorders fail during second year). The transition probability of state S2 to state Ss is 1-0.02 = 0.98.

As, the states Sw and Ss are absorbing states, the transition probability from state Sw to Sw is 1 and the transition probability from state Ss to Ss is 1.

One-step transition matrix is,

$P = \begin{bmatrix} States & S_1 &S_2 & S_w & S_s\\ S_1 & 0 & 0.97& 0.03& 0\\ S_2 & 0 & 0 & 0.02 &0.98 \\ S_w & 0 & 0 & 1 &0 \\ S_s & 0 & 0& 0 & 1 \end{bmatrix}$

(d)

Let X1, X2 be the states at the first and second transition. X0 is the initial state.

Probability that the manufacturer will have to honor the warranty

= Pr[X1 = Sw | X0 = S1] + Pr[X2 = Sw | X1 = S2, X0 = S1]

= 0.03 + 0.97 * 0.02 = 0.0494