Question

In Exercises 9–20, use the data in the following table, which lists drive-thru order accuracy at popular fast food chains (data from a QSR Drive-Thru Study). Assume that orders are randomly selected from those included in the table.

	McDonald’s	Burger King	Wendy’s	Taco Bell
Order Accurate	329	264	249	145
OrderNotAccurate	33	54	31	13

Fast Food Drive-Thru Accuracy If one order is selected, find the probability of getting an order that is not accurate or is from Wendy’s. Are the events of selecting an order that is not accurate and selecting an order from Wendy’s disjoint events?

Short answer

The probability of getting an inaccurate order or an order from Wendy’s is equal to 0.340.

No, these two events are not disjoint.

Step-by-step solution

Given information

Frequencies of food orders from four different chains are categorized under two subheadings, accurate and inaccurate orders.

Addition rule of probability

Let A and B be two events.

Theprobability of occurrence of either event A, B,or bothhas the following expression:

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

Here, denotes the probability of occurrence of A and B together. This probability needs to be subtracted under the addition rule.

Calculation

The following table shows all the subtotals and the grand total:

	McDonald’s	Burger King	Wendy’s	Taco Bell	Totals
Order Accurate	329	264	249	145	987
Order Not Accurate	33	54	31	13	131
Totals	362	318	280	158	1118

Let E be the event of getting an accurate food order.

Let F be the event of getting a food order from Wendy’s.

The total number of food orders is equal to 1,118.

The number of wrong food orders is calculated as shown below:

$33+54+31+13=131$

The number of food orders from Wendy’s is calculated as shown below:

$249+31=280$

The number of inaccurate food orders from Wendy’s is equal to 31.

So, the required probabilities are the following:

$$\begin{gathered} P\left(E\right)=\frac{131}{1118} \\ P\left(F\right)=\frac{280}{1118} \\ P\left(EandF\right)=\frac{31}{1118} \end{gathered}$$

Now, the probability (E or F) is as follows:

$$\begin{gathered} P\left(EorF\right)=P\left(E\right)+P\left(F\right)-P\left(EandF\right) \\ =\frac{131}{1118}+\frac{280}{1118}-\frac{31}{1118} \\ =\frac{380}{1118} \\ =0.340 \end{gathered}$$

Therefore, the probability of getting a wrong order or an order from Wendy’s is0.340.

Identifying the disjoint property

Two events are said to be disjoint if theycannot occur simultaneously.

Since the event of getting a food order from Wendy’s can occur simultaneously with the event of getting a wrong order, the given two events are not disjoint.

Mathematically, disjoint events E and F have the following probability 0.

$P\left(EandF\right)=0$

As, $P\left(EandF\right)=\frac{31}{1118}\left(\text{Non - zero}\right)$, the events are disjoint.