The answer is actually both the first and the third
are likely to be significantly greater than 1/2. The second is not.
Assuming the keys have as many 0s and 1s
as the good key rotor should be designed to have,
this is likely to be equal to 1/2.
There is no reason to expect it to be greater
than 1/2 any more likely than it'd be less than 1/2.
We're XORing two consecutive key bits.
The other two are different from 1/2.
This is what gives us the opportunity and what gave Bletchley Park the insight
needed to break the cipher.
For the third one, this followed from the structure of the machine.
When the S wheels advance this probability is about 1/2,
but when they don't advance ΔS is always 0.
This means the probability that ΔS is 0 is significantly greater than 1/2.
It turns out for the structure of the Lorenz machine it's about 0.73.
To know that you'd have to look in more detail at the structure of the M wheels
to know how frequently the S wheels advance.
When they don't advance we know the result is 0.
When they do advance about half the time the result will be 0.
Getting the first one right required a little more linguistic insight.
The reason this is greater than 1/2 depends on subsequent message letters.
If adjacent letters in the message are the same,
that ensures that ΔM is equal to 0.
It turns out that this is a property that many languages have.
You can see that English has it.
We have repeated letters here. We also have some repeated letters in "wheels."
It's more like that you would expect by just the normal letter distribution
for subsequent letters to be identical.
It turns out that this is a property of German that about 3.3% of digraphs,
meaning two adjacent letters are the same letter.
That means the probability that the messages are equal--
well, they could be equal for lots of other reasons,
but this bias is towards being more likely to be zero than non-zero.
It turns out that that's 0.61 probability for German.
Both of these are numbers that you didn't have enough information to guess on your own.
You would need to have a analysis of German text to know
that this is the probability here,
and you would need a lot more details on the M wheels to be able to get that.
Don't feel bad if you didn't get this quiz correct,
although the structure of the machine should have been enough to guess
that this is greater than 1/2,
and if you're familiar with German
or could guess that it has properties similar to English,
you might have been able to get this one as well.
