P$7.(20$ points$)$ A system's FIT $($failure$-$in$-$time$)$ rate is defined as the average number of failures observed in one billion hours. For a single component system with a failure rate $($MTTF of $\frac{1}{},$ FIT $=\frac{{10}^9}{?}$ MTTF$.$ Consider a soft error strike on a single SRAM bit cell in a single clock

cycle with a failure rate of $={10}^{-25}.$ Assume a cache has only one word of $64$ bits, and the word is fetched into the cache during the first program execution cycle and stays there.

$($a$)$ What is the probability of having a single$-$bit error caused in the cache during the execution of a program that runs for a billion cycles? This is called the intrinsic FIT rate.

$($b$)$ Consider a program where a cache line is written over every million cycles. In this case, what is the FIT rate of this cache?

$($c$)$ Consider the case when each cache line is protected by a single parity bit. Compute this cache's DUE $($detected but unrecoverable error$)$ FIT rate and SDC $($Silent data corruption, error

is not detected$)$ FIT rate.