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 from McDonald’s or an order that is accurate. Are the events of selecting an order from McDonald’s and selecting an accurate order disjoint events?

Short answer

The probability of getting an order from McDonald’s or an accurate order is equal to 0.9123.

No, these two events are not disjoint.

Step-by-step solution

Given information

The given data depicts the number of food orders from four different outlets under two categories, accurate order and inaccurate order, in a tabulated form.

Addition rule of probability

For two events, A and B, the probability is that either event A, B or both occur. It has the following expression:

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

Disjoint events

Two events are said to be disjoint if theyboth cannot occur together.

That is,

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

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 a food order from McDonald’s.

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

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

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

$329+33=362$

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

$329+264+249+145=987$

The number of accurate food orders from McDonald’s is 329.

So, the required probabilities are obtained as shown below:

$P\left(E\right)=\frac{362}{1118}$

$P\left(F\right)=\frac{987}{1118}$

$P\left(EandF\right)=\frac{329}{1118}$

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

$$\begin{gathered} P\left(EorF\right)=P\left(E\right)+P\left(F\right)-P\left(EandF\right) \\ =\frac{362}{1118}+\frac{987}{1118}-\frac{329}{1118} \\ =\frac{1020}{1118} \\ =0.9123 \end{gathered}$$

Therefore, the probability of getting a food order from McDonald’s or getting an accurate food order is equal to0.912.

Identifying the disjoint property

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

Mathematically,

$$\begin{gathered} P\left(EandF\right)=\frac{329}{1118} \\ \ne 0 \end{gathered}$$

Thus, the events are not disjoint.