# Homework #1: Counting things

Due: April 24, 2017 at 14:00.

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# Problems

## Problem 1

In class we showed that the distribution of sums of dots on a pair of dice is:

sum 2 3 4 5 6 7 8 9 10 11 12
frequency 1 2 3 4 5 6 5 4 3 2 1

Count the number of combinations of the number of dots on a pair of dice that achieves a particular:

1. product
2. difference
3. quotient

of the pair and make a frequency table for each arithmetic operation (a, b, and c).

Discussion:

The distribution of ordered pairs is said to be uniform. For each pair, the frequency seen in a series of throws of the dice should be about the same. Theoretically, this occurs because each (ordered) pair occurs once in a list of pairs (not shown here -- usually displayed as a square table with six rows and six columns). The frequency of sums is nonuniform (different frequencies for different values). Theoretically this difference occurs because there are often multiple pairs of dice that result in the same sum and the number of appropriate pairs varies according to the sum specified.

## Problem 2

Did you notice any interesting similarities or differences between the frequency distributions? Did you notice any interesting similarities or differences between the random variables (i.e., the function from the pairs of dice to the mathematical result)?

## Problem 3

Describe a model of "what is the gender of the first person to arrive in the classroom of 'Mathematics for Policy and Planning Science'" with an underlying set whose probabilities are uniform. What "model" means here is "What things do you count to determine the probability that the first person to arrive is female?"

Do you think this model actually describes the probability that the first person to arrive is female accurately? If so, why? If not, why not?

## Problem 4

Think of a practical or daily life application where you can explain observed rates of occurance with a model with an underlying uniform distribution, but a nonuniform distribution for the observed (or practically relevant) outcomes. What is the underlying set composed of? Why should its elements be uniformly distributed? What is the function relating the underlying things to observed outcomes?

## Problem 5

(Optional) Are there other ways to combine the two numbers on a pair of dice into an interesting single result with non-uniform probability? (Note: I don't have a good answer in mind as of writing this question, so don't lose sleep trying to answer it!)

## Problem 6

(Optional) Here's a brain-breaker for those of you who think you're good at math. The numbers on dice are all positive, and therefore it makes sense to take logarithms. There's a one-to-one relationship between numbers and their logarithms, so given a number we can find its logarithm, and it is the unique number with that logarithm. And given a number that is a logarithm there's a unique number it is the logarithm for.

Now, after taking logarithms, multiplication becomes addition. Therefore we might expect that there should be a one-to-one relationship between the distribution of sums of two dice and the distribution of products of two dice. That is wrong.

Explain why.

Due: April 25, 2015 at 14:00.