Chapter 8.5, Problem 27PPS

(a)

To determine

To rewrite: The formula to solve for the digit if given the probability.

Answer to Problem 27PPS

The formula to solve for the digit is $d=\frac{1}{{10}^P-1}$ .

Explanation of Solution

Given:

The given Benford formula that predicts the probability of a digit $d$ being the first digit of the set is given as:

  $P={\log }_{10}\left(1+\frac{1}{d}\right)$

Calculation:

To find the value $d$ is calculated as:

  $\begin{array}{c}P={\log }_{10}\left(1+\frac{1}{d}\right)\\ 1+\frac{1}{d}={10}^P\left[{\log }_{b}a=y\iff a=b^y\right]\\ \frac{1}{d}={10}^P-1\\ d=\frac{1}{{10}^P-1}\end{array}$

Therefore the formula to solve for the digit is $d=\frac{1}{{10}^P-1}$ .

(b)

To determine

To find: The digit that has a $9.7\%$ probability of being selected.

Answer to Problem 27PPS

The digit $4$ has a $9.7\%$ probability of being selected.

Explanation of Solution

To find the digit that has a $9.7\%$ probability of being selected, substitute $P=0.097$ in the formula $P={\log }_{10}\left(1+\frac{1}{d}\right)$

The calculation is given as:

  $\begin{array}{c}0.097={\log }_{10}\left(1+\frac{1}{d}\right)\\ 1+\frac{1}{d}={10}^{0.097}\left[{\log }_{b}a=y\iff a=b^y\right]\\ \frac{1}{d}=1.25-1\\ \frac{1}{d}=\frac{1}{4}\end{array}$

Further simplified as:

  $d=4$

Therefore the digit $4$ has a $9.7\%$ probability of being selected.

(c)

To determine

To find: The probability that the first digit is $1$ .

Answer to Problem 27PPS

The probability that the first digit is $1$ is $30.103\%$ .

Explanation of Solution

Given:

The value is given as:

  $\left({\log }_{10}2\approx 0.30103\right)$ .

Calculation:

To find the probability that the first digit is $1$ , substitute $d=1$ in the formula $P={\log }_{10}\left(1+\frac{1}{d}\right)$ is as:

  $$

  $\begin{array}{c}P={\log }_{10}\left(1+\frac{1}{1}\right)\\ P={\log }_{10}2\\ P=0.30103\\ P=30.103\%\end{array}$

Therefore the probability that the first digit is $1$ is $30.103\%$ .