TAOCP 3.1 Exercise 1
A suitable method for obtaining a decimal digit at random is one that produces each of the digits $0,1,\dots,9$ with equal probability and independently of prior choices.
Exercise 1. ▶ [20] Suppose that you wish to obtain a decimal digit at random, not using a computer. Which of the following methods would be suitable?
a) Open a telephone directory to a random place by sticking your finger in it somewhere, and use the units digit of the first number found on the selected page. b) Same as (a), but use the units digit of the page number. c) Roll a die that is in the shape of a regular icosahedron, whose twenty faces have been labeled with the digits 0, 0, 1, 1, …, 9, 9. Use the digit that appears on top, when the die comes to rest. (A felt-covered table with a hard surface is recommended for rolling dice.) d) Expose a geiger counter to a source of radioactivity for one minute (shielding yourself) and use the units digit of the resulting count. Assume that the geiger counter displays the number of counts in decimal notation, and that the count is initially zero. e) Glance at your wristwatch; and if the position of the second-hand is between $6n$ and $6(n+1)$ seconds, choose the digit $n$. f) Ask a friend to think of a random digit, and use the digit he names. g) Ask an enemy to think of a random digit, and use the digit he names. h) Assume that 10 horses are entered in a race and that you know nothing whatever about their qualifications. Assign to these horses the digits 0 to 9, in arbitrary fashion, and after the race use the winner's digit.
Verified: yes
Solve time: 1m06s
Solution
A suitable method for obtaining a decimal digit at random is one that produces each of the digits $0,1,\dots,9$ with equal probability and independently of prior choices. We examine each proposed method in turn.
(a) Opening a telephone directory to a random place by sticking a finger somewhere and using the units digit of the first number found on that page. The units digit of a telephone number is generally uniformly distributed among $0$ through $9$ because telephone numbers are assigned independently of each other, modulo administrative constraints. The selection of the page by a finger introduces a negligible bias provided that the placement is approximately uniform. Therefore, this method is suitable.
(b) Using the units digit of the page number. Page numbers in a telephone directory are consecutive integers, and consecutive integers modulo $10$ cycle through all digits $0$ through $9$ equally often. If the selection of the page is uniform, then each units digit occurs with frequency $1/10$. Hence, this method is also suitable.
(c) Rolling a 20-faced die with digits $0,0,1,1,\dots,9,9$. Each digit appears on exactly two faces of the 20-sided die. When the die is rolled, each face is equally likely to appear on top. Therefore, each digit occurs with probability $2/20 = 1/10$. This method is suitable.
(d) Using the units digit of a Geiger counter reading over one minute. Radioactive decay events are independent and approximately Poisson distributed. The units digit of a Poisson-distributed count modulo $10$ is approximately uniform if the expected count is sufficiently large, and the process is uncontaminated by deterministic biases. Therefore, this method is suitable in principle, though practical limitations (such as small counts or equipment bias) may reduce uniformity.
(e) Glancing at a wristwatch and mapping the second-hand position to digits $0$ through $9$. If the second-hand moves uniformly over the interval $[0,60)$ seconds, then the assignment of digits $n$ to intervals $[6n,6(n+1))$ produces a uniform selection. This method is suitable.
(f) Asking a friend to think of a digit. Human choices are generally influenced by patterns, preferences, and psychological biases. Consequently, the distribution is not uniform, and successive choices may be correlated. This method is not suitable.
(g) Asking an enemy to think of a digit. The same reasoning as in (f) applies: human choice is biased and predictable to some extent. This method is not suitable.
(h) Assigning digits $0$ through $9$ to horses in a race of unknown outcome and using the winning horse's digit. If the race is fair and the horses are indistinguishable in qualification, then the winner is equally likely to be any horse. Therefore, each digit occurs with probability $1/10$. This method is suitable.
Thus, the suitable methods are (a), (b), (c), (d), (e), and (h). Methods (f) and (g) are unsuitable because human choice is biased.
The final answer is:
$\boxed{\text{a, b, c, d, e, h}}$