Noisy reading from flash storage

Flash memories are a type of storage media which provide orders of magnitude faster access to data as compared to hard disks. However, one of the

downsides of flash memories is that they are prone to error when reading.

You have two flash memory devices, F$1$ and F$2.$ The noisy readings from

F$1$ and F$2$ are modeled as follows:

$$ F$1$: For any stored bit, the value read is flipped with probability $1.$

$$ F$2$: For any stored bit, the value read is flipped with probability $2.$

Suppose you write a bit into both F$1$ and F$2($i$.$e$.,$ the same bit is written

into both devices$),$ and that F$1$ and F$2$ act independently on that bit. A day

Mor Harchol$-$Balter. Introduction to Probability for Computing,

Cambridge University Press, $2024.$

$3\mathrm{.}4$ Exercises $57$

later, you read the bit that you wrote from F$1$ and from F$2.$ Represent the

value read from F$1$ by the r$.$v$.1$ and the value read from F$2$ by $2.$ Assume

that the stored bit is represented by $,$ where $$ is equally likely to be $0$ or

$1,$ barring any other information.

$($a$)$ Assume that $1=0\mathrm{.}1$ and $2=0\mathrm{.}2,$ that is$,$ the probability of flipping is low. Are $1$ and $2$ dependent? Explain using the definition of

independence of random variables.

$($b$)$ Repeat when $1=0\mathrm{.}5$ and $2=0\mathrm{.}2.$ Now are $1$ and $2$ dependent?

$($c$)$ Repeat when $1=0\mathrm{.}7$ and $2=0\mathrm{.}8.$ Now are $1$ and $2$ dependent?

$($d$)$ For what values of $1$ and $2$ do you conjecture that $1$ and $2$ are