Can you beat Piggy in this simple dice game? Can you figure out
Piggy's strategy, and is there a better one?
This tool allows you to create custom-specified random numbers,
such as the total on three dice.
Think that a coin toss is 50-50 heads or tails? Read on to appreciate the ever-changing and random nature of the world in which we live.
In 1881 an astronomer, Newcomb, first noticed a very bizarre property of some naturally occurring sets of numbers: if you list the surface areas of all the rivers in a country, about $30\%$ of them are numbers that have $1$ as a first digit, about $18\%$ have $2$ as a first digit and so on, with only about $5\%$ of them having $9$ as a first digit.What's more, if you convert the lengths into any other unit (miles, feet, mm, etc) the distribution of first digits remains the same (we say, the distribution
is 'scale invariant'.) The same pattern of first digits, occurs in many sets of seemingly random numbers. It is called Benford's Law, after its second discoverer physicist Frank Benford, working in 1938. In this problem we shall use probability to predict the numbers observed by Newcomb.
You will need to know that a function f(x) is called 'scale invariant' if scaling x by a fixed amount does not change the shape of the function. Mathematically, the property of scale invariance is written as: f(Ax) = k f(x) for fixed numbers A and k
Show that if a probability density function $f(x)$ with $x> 0$ is scale invariant then
$f(Ax) = f(x) / A$
Can a function $f(x)$ be both scale invariant and a probability density function if $x$ is allowed to take any non-negative value? Experiment with various forms of $f(x)$ to try to find out.
How would your results change if $f(x)$ was restricted to take values a< x< b, for some positive numbers $a$ and $b$?
Suppose that $a = 1$ and $b = 1,000,000$. Which of the functions will make a scale invariant probability density function? For this density, show that $$P(1< x< 2) = P(100< x< 200) =P(1000000< x< 2000000)$$ Suppose that a number X is drawn randomly from this distribution. Calculate the probability that its first digit is$1$. Extend this to calculate the probability that the
first digit is $2, 3, 4, ...., 9$. How would these results change if $b$ was $1,000,000,000$ or $1,000,000,000,000$?