Problem of the Week Day 5: Week of 6/17/13 - 6/21/13

Today is the final day of the first 2013 problem of the week! The goal of the problem is to find x, so I have saved the x variable for the last part. In the hard problem, we used x with a subscript, but the plain x will represent the solution to today's problem. We have already covered trigonometry, algebra, number theory, and geometry, so I thought it would be good to devote Friday to probability. I might incorporate some game theory as well on Fridays in later problems.

Easy:

Take the following set of data:

59, 38, 45, p, 29, g, 27, y + 1.24, q, 41, z + 5.92, 37, b, 21

What is the median of this sequence?

x = _____

Medium:
First, solve for the following six variables, which will come up in the data set you will be analyzing.

d = a ÷ 1000
g = h ÷ 7
j = u ÷ 2
k = u ÷ 4
r = p ÷ 10
s = q ÷ 3

The following data represents the correlation between the age of test subjects (x) and the average dollar amount that they spend at a nearby shopping mall per visit (y). Some of the data was rounded by the data collectors, and some of it is as accurate as possible, meaning that there will be some longer decimals.

x	y
s	q + r + g
t	u
q - h	q
h - u	h - t
r	h + t
r - t	h - r
d + t	p - h
g	h - g
d	h
2t	q + t
k	q + t
k	u + t
j	4s

Find the function that describes the line of best fit for this data set. Use the traditional y = mx + b to make the equation of the line, using m and b as your variables. Then, determine the average spending of a 65-year-old woman using the equation. The average spending will be the x value that we always end the problem with.

m = ______
b = ______
x = ______

Hard:
First, use the variables throughout the week to solve for m.

        t – 0.77 – p(100q₁₀ + q₁₂)(q₉ – q₁₂)
m = –––––––––––––––––––––––––––––––
       q₀ + 2c – a + b + q₆ + q₁₀ + q₁₁ + q₁₄

Now, pretend that there is a room with m people inside of it. Determine what the odds are that in this room, two people share the same birthday (just the month and date, not the year). In other words, solve the birthday paradox for a room of m people. x will represent the odds of this happening.

x = _____%