Probability of failure of a light bulb in years

markov chainsprobabilitytransition matrix

Let's assume we have a lightbulb with a maximum lifespan 4 years. We are asked to create a transition matrix (Markov chain theory) for the bulb. The bulb is checked once a year and if it's found that the bulb does not work it is replaced by a new one.

We know that the probabilities of failure during the 4 year period are: 0.2, 0.4, 0.3 and 0.1. After four years, the lightbulb is replaced with probability 1. So we have the following 4 states:

  • $S_0$ – the lightbulb is new
  • $S_1$ – the lightbulb is 1 year old
  • $S_2$ – the lightbulb is 2 years old
  • $S_3$ – the lightbulb is 3 years old

How to create the probability transition matrix? For the first year, it's clear. The bulb goes dead with probability $p_{0,0} = 0.2$ and it keeps working with probability $p_{0,1} = 0.8$. But I am not sure how to calculate to the following years. In the materials for my course, I found the following calculation:

$$
p_{2,1} = \frac{0.4}{0.8} \, , p_{2,3} = \frac{0.3+0.1}{0.8} = 0.5 \, , p_{3,1} = \frac{0.3}{0.4} = 0.75 \, , p_{3,4} = \frac{0.1}{0.4} = 0.25
$$

So the probability transition matrix is:

\begin{bmatrix}
0.2&0.8&0&0\\0.5&0&0.5&0\\0.75&0&0&0.25\\1&0&0&0\end{bmatrix}

Is this correct? I fail to see why $p_{2,3}$ uses the probability of failure in the last year.

Best Answer

The probabilities known, summing to $1$, are the probability at birth of failing during the 1st, 2nd, 3d or 4th year. Upon failure, the bulb is replaced with a new one, which thus has the same probabilities as above.
We have therefore the following scheme.

Light_Bulb_1

So the probability (at birth) $P_2$ to fail in the 2nd year (not before, and not after) will be given by the probability to survive for the first year times the probability $p_2$ to fail exactly in the 2nd year (given that it survived the first). And analogously for the others, i.e. $$ \eqalign{ & p_{\,1} = 0.2 \cr & \left( {1 - p_{\,1} } \right)p_{\,2} = 0.8 \cdot p_{\,2} = P_{\,2} = 0.4\quad \Rightarrow \quad p_{\,2} = 0.5 \cr & \left( {1 - p_{\,1} } \right)\left( {1 - p_{\,2} } \right)p_{\,3} = P_{\,3} \quad \quad \Rightarrow \quad p_{\,3} = 0.75 \cr & \left( {1 - p_{\,1} } \right)\left( {1 - p_{\,2} } \right)\left( {1 - p_{\,3} } \right)p_{\,4} = P_{\,4} \quad \Rightarrow \quad p_{\,4} = 1 \cr} $$

And $p_k,(1-p_k)$ are the entries of the matrix.

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