# Expert Answers

1. For each of the following figures, decide whether or not the figure shows a valid probability distribution, and if so, identify it as a uniform or non-uniform distribution.

2. Consider a five-sided die.

(a)The set of possible values for the die roll is the integers from 1 to 5. Draw the probability distribution graph for the set of these possible values, and label the values on the axes.

(b) Is this a uniform distribution over the set of possible values?

3. Now consider two five-sided dice.

(a)If you were to roll the two dice, what would be the possible values for the sum of the two dice?

(b) Draw the probability distribution graph for the set of possible values, and label the values on the axes. Don’t put any values on the graph that have a probability of 0.

(c)Is this a uniform distribution over the set you gave in part (a)?

4. Now consider the following table for an encryption function designed to be used for encrypting either a 0 or a 1.

(a) Imagine an experiment in which the key is chosen uniformly at random. Let A be the random variable that is the encryption of plaintext 0. Draw the probability distribution of A.

(b) Let B be the random variable that is the encryption of plaintext 1. Draw the probability distribution of B.

5. Consider the following table for a decryption function:

(a) Fill in the arrows in the following function diagrams to represent the specializations of the above decryption function:

(b) Let the key be selected randomly according to a uniform distribution on {0,1,2}.

i. Let A be the decryption of the cyphertext 0 using the random key. Draw the probability distribution of A.

ii. Let B be the decryption of the cyphertext 1. Draw the probability distribution of B.

iii. Let C be the decryption of the cyphertext 2. Draw the probability distribution of C.

2. Consider a five-sided die.

(a)The set of possible values for the die roll is the integers from 1 to 5. Draw the probability distribution graph for the set of these possible values, and label the values on the axes.

(b) Is this a uniform distribution over the set of possible values?

3. Now consider two five-sided dice.

(a)If you were to roll the two dice, what would be the possible values for the sum of the two dice?

(b) Draw the probability distribution graph for the set of possible values, and label the values on the axes. Don’t put any values on the graph that have a probability of 0.

(c)Is this a uniform distribution over the set you gave in part (a)?

4. Now consider the following table for an encryption function designed to be used for encrypting either a 0 or a 1.

(a) Imagine an experiment in which the key is chosen uniformly at random. Let A be the random variable that is the encryption of plaintext 0. Draw the probability distribution of A.

(b) Let B be the random variable that is the encryption of plaintext 1. Draw the probability distribution of B.

5. Consider the following table for a decryption function:

(a) Fill in the arrows in the following function diagrams to represent the specializations of the above decryption function:

(b) Let the key be selected randomly according to a uniform distribution on {0,1,2}.

i. Let A be the decryption of the cyphertext 0 using the random key. Draw the probability distribution of A.

ii. Let B be the decryption of the cyphertext 1. Draw the probability distribution of B.

iii. Let C be the decryption of the cyphertext 2. Draw the probability distribution of C.

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