Question

use the data in the following table, which lists drive-thru order accuracy at popular fast food chains. assume that orders are randomly selected from those included in the table.

drive-thru restaurant
\begin{tabular}{|l|c|c|c|c|}
hline & a & b & c & d \\
hline order accurate & 335 & 275 & 230 & 122 \\
hline order not accurate & 35 & 60 & 31 & 11 \\
hlineend{tabular}

if two orders are selected, find the probability that they are both accurate. complete parts (a) and (b) below.

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

the probability is 0.7674. the events are independent.
(do not round until the final answer. round to four decimal places as needed.)

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

the probability is \boxed{} the events \dropdown{} independent
(do not round until the final answer. round to four decimal places as needed.)

Explanation:

Step1: Calculate total orders

First, find the total number of orders. For accurate orders: \(335 + 275 + 236 + 122 = 968\). For not accurate: \(35 + 60 + 31 + 11 = 137\). Total orders \(N = 968 + 137 = 1105\).

Step2: Probability without replacement

When selecting without replacement, the probability that the first order is accurate is \(P_1=\frac{968}{1105}\). After selecting one accurate order, the number of accurate orders left is \(967\) and total orders left is \(1104\). So the probability the second is accurate is \(P_2=\frac{967}{1104}\). The combined probability is \(P = P_1\times P_2=\frac{968}{1105}\times\frac{967}{1104}\).
Calculate \(\frac{968\times967}{1105\times1104}=\frac{936056}{1220920}\approx0.7667\).
For independence: When sampling without replacement, the outcome of the first selection affects the second (since the total and number of accurate orders change), so events are not independent.

Answer:

The probability is \(0.7667\). The events are not independent.