Question

Based on previous records, 17% of the vehicles passing through a tollbooth have out-ofstate plates. A bored tollbooth worker decides to pass the time by counting how many vehicles pass through until he sees two with out-of-state plates. We would like to perform a simulation to estimate the average number of vehicles it takes to find two with out-of-state plates.30

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

b. Perform 3 trials of the simulation using the random digits given here. Copy the digits onto your paper and mark directly on or above them so that someone can follow what you did.

Short answer

Part a) 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.

Part b) Then we note that we'll need $13$vehicles until we get two with out-of-state plates.

Step-by-step solution

Part a) Step 1: Explanation

In the question, it is stated that 17% of the vehicles passing through the tollbooth have out-of-state plates, but we are only interested in the number of vehicles that pass through until two out-of-state plates are observed.

We have,

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

We'll now choose a row from the table of random numbers. The first two-digit number will be chosen after that. If the digit is between $00$and $16$(inclusive), the vehicle has out-of-state plates; otherwise, the vehicle does not have out-of-state plates.

This process will be repeated until two vehicles with out-of-state plates have been obtained. Then we'll keep track of how many trials are required until two vehicles have out-of-state license plates.

Part b) Step 1: Given information

First trial: require three vehicles.
The second trial: required$14$vehicles.
The third trial: required $13$vehicles.

Part b) Step 2: Explanation

In the question, it is stated that 17% of the vehicles passing through the tollbooth have out-of-state plates, whereas we are interested in the total number of vehicles passing through until two vehicles with out-of-state plates are seen.

We have,

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

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

We'll start with the first trial now.

The first two-digit number will be chosen. The vehicle has out-of-state plates if the digit is between $00$and $16$(inclusive); otherwise, the vehicle does not have out-of-state plates. This process will be repeated until two vehicles with out-of-state plates have been obtained.

$41=$Vehicles do not have out-of-state plates

$05\Rightarrow$ Vehicles have out-of-state plates
$09\Rightarrow$Vehicles have out-of-state plates

Then we note that we'll need three vehicles until we can get two with out-of-state license plates.

We will now conduct a second trial as follows:

The first two-digit number will be chosen. The vehicle has out-of-state plates if the digit is between $00$and $16$(inclusive); otherwise, the vehicle does not have out-of-state plates. This process will be repeated until two vehicles with out-of-state plates have been obtained.

$20\Rightarrow$Vehicles do not have out-of-state plates
$31\Rightarrow$ Vehicles do not have out-of-state plates
$06\Rightarrow$ Vehicles have out-of-state plates
$44\Rightarrow$ Vehicles do not have out-of-state plates
$90\Rightarrow$ Vehicles do not have out-of-state plates
$50\Rightarrow$ Vehicles do not have out-of-state plates
$59\Rightarrow$ Vehicles do not have out-of-state plates
59⇒Vehicles do not have out-of-state plates
$88\Rightarrow$Vehicles do not have out-of-state plates
$43\Rightarrow$ Vehicles do not have out-of-state plates
$18\Rightarrow$ Vehicles do not have out-of-state plates
$80\Rightarrow$ Vehicles do not have out-of-state plates
$53\Rightarrow$ Vehicles do not have out-of-state plates
$11\Rightarrow$Vehicles have out-of-state plates

Then, until we get two vehicles with out-of-state plates, we'll need $14$vehicles.

We will now proceed to the third trial, as follows:

The first two-digit number will be chosen. The vehicle has out-of-state plates if the digit is between $00$and $16$(inclusive); otherwise, the vehicle does not have out-of-state plates. This process will be repeated until two vehicles with out-of-state plates have been obtained.

$58\Rightarrow$Vehicles do not have out-of-state plates
44⇒Vehicles do not have out-of-state plates
69⇒Vehicles do not have out-of-state plates
94⇒Vehicles do not have out-of-state plates
86⇒Vehicles do not have out-of-state plates
85⇒Vehicles do not have out-of-state plates
79⇒Vehicles do not have out-of-state plates
67⇒Vehicles do not have out-of-state plates
05⇒Vehicles have out-of-state plates
81⟹Vehicles do not have out-of-state plates
18⇒Vehicles do not have out-of-state plates
45⇒Vehicles do not have out-of-state plates
14⇒ Vehicles have out-of-state plates

Then we note that we'll need $13$vehicles until we get two with out-of-state plates.