Suppose that on the planet Binary, the written language uses an alphabet that contains only

two letters X and Y$.$ Also, suppose that in the Binarian language, the letter X occurs $75\%$ of

the time, while Y occurs $25\%$ of the time. Finally, assume that you have two messages in the

Binary language, and the messages are of equal length $[15$ points$].$

$1$

a$.$ If you compare the corresponding letters of the two messages, what fraction of the time

will the letters match?

b$.$ Suppose that one of the two messages is encrypted with a simple substitution, where X is

encrypted as Y and Y is encrypted as X$.$ If you now compare the corresponding letters of

the two messages$$one encrypted and one not$$what fraction of the time will the letters

match?

c$.$ Suppose that both of the messages are encrypted with a simple substitution, where X is

encrypted as Y and Y is encrypted as X$.$ If you now compare the corresponding letters

of the two messages$$both of which are encrypted$$what fraction of the time will the

letters match?

d$.$ Suppose instead that you are given two randomly generated $$messages$$ that use only

the two letters X and Y$.$ If you compare the corresponding letters of the two messages,

what fraction of the time will the letters match