Using Benford's law According to Benford's law (Exercise 15, page 377), the probability that the first digit of the amount of a randomly chosen invoice is an 8 or a 9 is 0.097. Suppose you examine randomly selected invoices from a vendor until you find one whose amount begins with an 8 or a 9 .

a. How many invoices do you expect to examine before finding one that begins with an 8 or 9 ?

b. In fact, the first invoice you find with an amount that starts with an 8 or 9 is the 40 th invoice. Does this result provide convincing evidence that the invoice amounts are not genuine? Calculate an appropriate probability to support your answer.

Given,$p=0.097$

Formula used:

The expected value

$\mu =\frac{1}{p}$

The number of independent trials required until first success is distributed geometrically.

A geometric distribution's expected value is

$\mu =\frac{1}{p}=\frac{1}{0.097}=10.3039$

Hence, it is expected that to examine about $10.3093$invoices until you achieve your first success, which is an invoice starting with an 8 or 9.

Given$p=0.097$

Formuale to be used

Geometric probability:

$P(X=k)=q^{k-1}p=(1-p{)}^{k-1}p$

Addition rule:

$P(A\cup B)=P\left(A\text{or}B\right)=P\left(A\right)+P\left(B\right)$

Complement rule:

$P\left(A°\right)=P(notA)=1-P\left(A\right)$

Consider,$p=0.097$

Compute the binomial probability definition at $k=1,2$, and $3\dots 40$:

$P(X=1)=(1-0.097{)}^{1-1}(0.097)=0.097$

Similarly

localid="1654759250726" $$\begin{gathered} P(X=2)=0.0876 \\ P(X=3)=0.0791 \\ P(X=39)=0.0020 \\ P(X\le 39)=P(X=1)+P(X=2)+P(X=3)+\dots +P(X=39) \\ =0.097+0.0876+0.0791+\dots +0.0020 \\ =0.9813 \end{gathered}$$

Using the complement rule:

$$\begin{gathered} P(X\ge 40)=1-P(X\le 39) \\ =1-0.9813 \\ =0.0187=1.87\% \end{gathered}$$

Using Ti83/84- calculator

It has been discovered that the likelihood of achieving the first success on the ${40}^{\text{th}}$invoice or after it is small (less than $0.05$), This demonstrates that the occurrence is unlikely to occur, and hence there is sufficient convincing evidence that the invoice amounts are not genuine.