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 two orders are selected, find the probability that both of them are not accurate.

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?

Short answer

a. The probability that both are inaccurate orders selected with replacement is equal to 0.0137.

Yes, the events are independent.

b. The probability that both are inaccurate orders selected without replacement is equal to 0.0136.

No, the events are not independent.

Step-by-step solution

Given information

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

Multiplication rule of probability

Let A and B be two events.

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

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

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

Independence of events

• If selections are made with replacement, events are independent.
• If selections are made without replacement, events are not independent.

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

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

The number of inaccurate food orders is equal to 131.

The probability of selecting an inaccurate food order is as follows:

$P\left(\text{inaccurate}\text{order}\right)=\frac{131}{1118}$

Let E be the event of selecting an inaccurate order on the first try.

Let F be the event of selecting an inaccurate order on the second try.

a.

As selections are made with replacement, the total number of orders and the number of inaccurate orders remains the same for both tries.

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

The probability that both the orders are inaccurate is as follows:

$$\begin{gathered} P\left(EandF\right)=P\left(E\right)\times P\left(F|E\right) \\ =\frac{131}{1118}\times \frac{131}{1118} \\ =0.0137 \end{gathered}$$

Therefore, the probability of selecting both inaccurate orders with replacement is equal to 0.0137.

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 inaccurate orders will decrease by one for the second try.

$P\left(F|E\right)=\frac{130}{1117}$

The probability that both the orders selected are inaccurate is as follows:

$$\begin{gathered} P\left(E\text{and}F\right)=P\left(E\right)\times P\left(F|E\right) \\ =\frac{131}{1118}\times \frac{130}{1117} \\ =0.0136 \end{gathered}$$

Therefore, the probability of selecting both inaccurate orders without replacement is equal to 0.0136.

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