Chapter 5, Problem T5.14SPT

(a)

To determine

To describe how you would use a table of random digits to perform the simulation.

Explanation of Solution

In the question it is given that $17\%$ of the vehicles passing the tollbooth have out-of-state plates, while we are interested in the number of the vehicles that pass through until two vehicles with out-of-state plates are observed. Thus, we have,

  $\begin{array}{l}17\%=0.17\\ =\frac{17}{100}\end{array}$

Now, we will choose a row from the random digits table. Then, we will select the first two-digit number. If the digit is between $00\text{ and 16}$ (including) then the vehicle has out-of-state plates else the vehicle does not have out-of-state plates.

This process will be repeated until we have obtained two vehicles that have out-of-state plates. Then we will keep the track of the number of trials required until two vehicles had out-of-state plates.

(b)

To determine

To perform three trials of simulations using the random digits given in the question.

Answer to Problem T5.14SPT

First trial: require three vehicles.

Second trial: required $14$ vehicles.

Third trial: required $13$ vehicles.

Explanation of Solution

In the question it is given that $17\%$ of the vehicles passing the tollbooth have out-of-state plates, while we are interested in the number of the vehicles that pass through until two vehicles with out-of-state plates are observed. Thus, we have,

  $\begin{array}{l}17\%=0.17\\ =\frac{17}{100}\end{array}$

Thus, the given rows from the random digits given in the tables are as:

  $\begin{array}{l}41050,92031,06449,05059,59884,31880,53115,\\ 84469,94868,57967,05811,84514,84177,06757,\\ 17613,15582,51506,81435\end{array}$

Now, we will do first trial,

We will select the first two-digit number. If the digit is between $00\text{ and 16}$ (including) then the vehicle has out-of-state plates else the vehicle does not have out-of-state plates. This process will be repeated until we have obtained two vehicles that have out-of-state plates.

  $\begin{array}{l}41\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ \text{05}\Rightarrow \text{ Vehicles has out-of-state plates}\\ \text{09}\Rightarrow \text{ Vehicles has out-of-state plates}\end{array}$

Then, we note that we require three vehicles until we obtain two vehicles with out-of-state plates.

Now, we will do second trial as,

We will select the first two-digit number. If the digit is between $00\text{ and 16}$ (including) then the vehicle has out-of-state plates else the vehicle does not have out-of-state plates. This process will be repeated until we have obtained two vehicles that have out-of-state plates.

  $\begin{array}{l}20\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 31\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ \text{06}\Rightarrow \text{ Vehicles has out-of-state plates}\\ \text{44}\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 90\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 50\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 59\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 59\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 88\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 43\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 18\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 80\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 53\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 11\Rightarrow \text{ Vehicles has out-of-state plates}\end{array}$

We then note that we required $14$ vehicles until we obtain two vehicle with out-of-state plates.

Now, we will do third trial as,

We will select the first two-digit number. If the digit is between $00\text{ and 16}$ (including) then the vehicle has out-of-state plates else the vehicle does not have out-of-state plates. This process will be repeated until we have obtained two vehicles that have out-of-state plates.

  $\begin{array}{l}58\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 44\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 69\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 94\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 86\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 85\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 79\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 67\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 05\Rightarrow \text{ Vehicles has out-of-state plates}\\ \text{81}\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 18\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 45\Rightarrow \text{ Vehicles does not have out-of-state plates}\\ 14\Rightarrow \text{ Vehicles has out-of-state plates}\end{array}$

We then note that we required $13$ vehicles until we obtain two vehicles with out-of-state plates.