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Imagine a distribution center where packages are sent to various locations. We want to know the likelihood that any random package passes through different branches of the delivery system by using the concept of the law of total probability. This law helps to simplify calculations involving composite events. It combines probabilities of several mutually exclusive events to determine the probability of a specific outcome.

In our context of a computer disk storage device, we are trying to find the probability mass function of the number of tracks over which the access arm passes. First, we determine the conditional probability of moving across a specific number of tracks given the starting track. However, the scenario doesn't stop there. By multiplying these conditional probabilities by the probabilities of being on each starting track and summing over them, we use the law of total probability.

This approach ensures that all possible paths the arm might take are accounted for. Think of the arm calculating the total possible paths and then integrating them into one holistic probability. It’s like reading a complete map rather than just understanding each possible route in isolation.

Conditional probability is crucial when dealing with the probability mass function (PMF) of how many tracks the arm crosses when moving from one request to another on a disk.

Let’s consider the scenario where you need to move from one bus stop to another but only have information regarding how likely you are to transition between specific pairs of spots. Conditional probability helps because it gives the possibility of one event happening provided another event has already occurred.

In the exercise, if the arm starts at track \(i\), it must learn how likely it is to end at track \(i+j+1\) or \(i-j-1\) to move across \(j\) tracks. The conditional probability \(P(X = j \mid T_1 = i)\) is pivotal as it evaluates the scenario of moving from the specific track \(i\) and landing at a track either through moving forwards or backwards, and how likely that move happens. This allows us to understand the tendencies of the arm’s movement based on its starting position. It’s like knowing the probability of you getting a certain grade given that you've studied a set number of hours.

In our storage device scenario, the notion of independent events is used to make calculations possible and simplify the reasonings. Think of independent events as flipping a coin twice where the first flip does not affect the outcome of the second flip. In this case, the track requests are independent.

This means if the arm goes to a track during the first request, it does not influence where it will go next. Each track request is random and doesn't change the likelihood of subsequent tracks being requested.

For the disk storage device exercise, because successive track accesses are independent, this greatly simplifies the process to compute the PMF for the number of tracks \(X\). It allows us to treat each track access in isolation without worrying about past sequences or path dependencies. Understanding this independence was critical because it meant we did not have to account for the sequences and could treat the probabilities as standalone, making the computations practical and feasible.