Final answer:
The probability that a file is lost due to machine failure, when it is stored on k machines with each having a failure probability of p, is p raised to the power of k (p^k). This calculation assumes that failures of different machines are independent events.
Step-by-step explanation:
To find the probability that the file is lost due to machine failure, considering there are no replicas and k blocks with each being on a different machine, we can use the concept of independent probabilities. Since machine failures are independent, the probability that all machines fail (and thus, the file is lost), is the product of each machine's probability of failure. As the problem states, the probability of a single machine failing is p.
Since we want all k blocks to be simultaneously unavailable due to machine failure, such an event would require each of the k machines to fail. The probability for the file to be lost is thus p raised to the power of k (p^k), representing the compounded probability of failure across all k machines. This formula is based on the rule that for independent events, the joint probability is the product of the probabilities of each event occurring.
In formal terms, if we define the event of losing the file as a 'success' (an unfortunate terminology in this context, but consistent with statistical definitions), we can state that the probability of success is p^k, wherein each trial (each machine's operation) is independent, and the failure of each contributes to the overall 'success' of losing the file.