Chapter 6.3, Problem 111E

(a)

To determine

To find: the invoices expected to examine before finding one that start with an 8 or 9.

Answer to Problem 111E

10.3093 invoices

Explanation of Solution

Given:

  $p=0.097$

Formula used:

The expected value

  $\mu =\frac{1}{p}$

Calculation:

The number of independent trials needed until first success follows a geometric distribution.

The expected value of a geometric distribution is

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

Therefore it is expected that to examine about 10.3093 invoices until get the first success which is an invoice that begins with an 8 or 9.

(b)

To determine

To Calculate: an appropriate probability to help the answer.

Answer to Problem 111E

Yes

Explanation of Solution

Given:

  $p=0.097$

Formula used:

Geometric probability:

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

Addition rule:

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

Complement rule:

  $P\left(A^{\circ }\right)=P\left(\text{not }A\right)=1-P\left(A\right)$

Calculation:

  $p=0.097$

Calculate the definition of binomial probability at k = 1, 2, and 3… 40:

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

Similarly

  $\begin{array}{l}P\left(X=2\right)=0.0876\\ P\left(X=3\right)=0.0791\end{array}$

  $P\left(X=39\right)=0.0020$

  $\begin{array}{c}p\left(X\le 39\right)=P\left(X=1\right)+P\left(X=2\right)+P\left(X=3\right)+\mathrm{...}+P\left(X=39\right)\\ =0.097+0.0876+0.0791+\mathrm{...}+0.0020\\ =0.9813\end{array}$

Using the complement rule:

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

Using Ti83/84- calculator

It is observed that the probability of getting the first success on the 40^th invoice or after it is small (less than 0.05), which shows that the event is not likely to get and therefore there is enough convincing proof that the invoice amounts are not genuine.