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 two orders are selected, find the probability that they are both from Taco Bell.

a. Assume that the selections are made with replacement. Are the events independent?

b. Assume that the selections are made without replacement. Are the events independent?

Frequencies of food orders of two types, accurate and inaccurate, are tabulated for four different fast-food chains.

Let A and B be two events.

Theprobability of occurrence of events A and B togetherhas the following notation:

$P\left(A\text{and}B\right)=P\left(A\right)\times P\left(B|A\right)$

Here, $P\left(B|A\right)$stands for the probability of B given A has already occurred.

Events under the multiplication can be independent or not based on the following situations:

• If selections are made with replacement, each of the several events can be considered independent.
• If selections are made without replacement, each of the several events can be considered dependent or not independent.

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

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

The number of food orders from Taco Bell is equal to 158.

The probability of selecting a food order from Taco Bell is as follows:

$P\left(\text{order}\text{from}\text{Taco}\text{Bell}\right)=\frac{158}{1118}$

Define events:

Let E be the event of selecting an order from Taco Bell on the first try.

Let F be the event of selecting an order from Taco Bell on the second try.

a.

Selections made with replacement, the total number of orders and the number of Taco Bell orders remains the same for both the selections:

$$\begin{gathered} P\left(E\right)=P\left(F|E\right) \\ =\frac{158}{1118} \end{gathered}$$

The probability that both the orders selected are from Taco Bell is as follows:

$$\begin{gathered} P\left(E\text{and}F\right)=P\left(E\right)\times P\left(F|E\right) \\ =\frac{158}{1118}\times \frac{158}{1118} \\ =0.0200 \end{gathered}$$

Therefore, the probability of selecting both the orders from Taco Bell with replacement is equal to0.0200.

As the selections are made with replacement, they are independent.

b.

As selections are made without replacement, the total number of orders and the number of Taco Bell orders will decrease by one for the second selection.

$P\left(F|E\right)=\frac{157}{1117}$

The probability that both the orders selected are from Taco Bell is as follows:

$$\begin{gathered} P\left(E\text{and}F\right)=P\left(E\right)\times P\left(F|E\right) \\ =\frac{158}{1118}\times \frac{157}{1117} \\ =0.0199 \end{gathered}$$

Therefore, the probability of selecting both the orders from Taco Bell without replacement is equal to0.0199.

As the selections are made without replacement, they are not independent.