Question

Consider the first 25 digits in the decimal expansion of π (3, 1, 4, 1, 5, 9, . . .).

(a) If you selected one number at random, from this set, what are the probabilities of getting each of the 10 digits?

(b) What is the most probable digit? What is the median digit? What is the average value?

(c) Find the standard deviation for this distribution.

Short answer

1. Probability of choosing each digit is shown in the solution.
2. Most probable digit = 3, Median digit = 4, Average value = 28.4.
3. $\mathrm{\sigma }=2.47$
   

Step-by-step solution

Solution:

First 25 decimal digits of π is,

$\mathrm{\pi }=3.141592653589793238462643...$

Finding the probability of choosing a particular digit

Part (a)

The probability of choosing a particular digit is the frequency, divided by the total number of digits

$\begin{array}{cc}0:\frac{0}{25}=0& 5:\frac{3}{25}=0.12\\ 1:\frac{2}{25}=0.08& 6:\frac{3}{25}=0.12\\ 2:\frac{3}{25}=0.12& 7:\frac{1}{25}=0.04\\ 3:\frac{5}{25}=0.20& 8:\frac{2}{25}=0.08\\ 4:\frac{3}{25}=0.12& 9:\frac{3}{25}=0.12\end{array}$

Write the set of all the numbers from smallest to largest

Part (b)

$\{1,1,2,2,2,3,3,3,3,3,4,4,4,5,5,5,6,6,6,7,8,8,9,9,9\}$

The highlighted number in this set represents the median

Median = 4

Most probable digit chosen is 3 since it’s the number with highest frequency

Calculating the average value

The average value is given by the expectation value.

Thus, the most probable digit, median, and average value are 4, 3, and 28.4 respectively.

Calculating the standard deviation

Standard deviation is given by

$\begin{array}{l}\mathrm{\sigma }=\sqrt{⟨{\mathrm{j}}^2⟩-⟨\mathrm{j}{⟩}^2}\\ \mathrm{\sigma }=\sqrt{28.4-4{.72}^2}\\ \mathrm{\sigma }=2.47\end{array}$

Thus, the standard deviation is 2.47.