1. (10 pts.) Suppose you have a language with only 3 letters: A, B, and C, which occur with frequencies 0.7, 0.2, and 0.1, respe
ctively. The following ciphertext was encrypted by the Vigen`ere method: BABABCBCAC. Assume that the key length is either 1, 2, or 3. Find the most likely key length and determine the most likely key using the method from class. Solution: We compare BABABCBCAC with its shifts by 1, 2, and 3, looking for overlaps. There are no overlaps for a shift of 1 or 3, but there are 6 overlaps for a shift of 2. Thus we expect the key length to be 2. Looking at every other letter, we see BBBBA and AACCC for the letters in even and odd positions, respectively. Due to the overwhelming frequency of the letter A in the language, we can reasonably assume that B decrypts to to A in the first sequence and C decrypts to A in the second. This gives us the key BC.
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Answer:
0.3844 = 38.44% probability that two independently surveyed voters would both be Democrats
Step-by-step explanation:
For each voter, there are only two possible outcomes. Either the voter is a Democrat, or he is not. The probability of the voter being a Democrat is independent of other voters. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
In which is the number of different combinations of x objects from a set of n elements, given by the following formula.
And p is the probability of X happening.
62% of the voters are Democrats
This means that
(a) What is the probability that two independently surveyed voters would both be Democrats?
This is P(X = 2) when n = 2. So
0.3844 = 38.44% probability that two independently surveyed voters would both be Democrats
All exercises involve the same concept, so I'll show you how to do the first, then you can apply the exact same logic to all the others.
The first thing you need to know is that, when a certain quantity multiplies a parenthesis, you can distribute that number to every element in the parenthesis. This means that
So, is multiplying the parenthesis involving
and
, and we distributed it:
multiplies both
and
in the final result.
Secondly, you have to know how to recognize like terms, because they are the only terms you can sum. Two terms can be summed if they have the same literal expression. So, for example, you cannot sum , and neither
exponents count.
But you can su, for example,
or
So, take for example exercise 9:
We distribute the 1.2 through the first parenthesis:
And you can distribute the negative sign through the second parenthesis (it counts as a -1 to distribute):
So, the expression becomes
Now sum like terms: