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

Explanation:

Step1: Calculate total number of orders

First, find the total number of orders (accurate and not - accurate).
Total number of accurate orders: $332 + 268+250 + 146=996$
Total number of not - accurate orders: $36 + 53+31 + 18 = 138$
Total number of all orders: $996+138 = 1134$

Step2: Calculate probability of one accurate order

The probability of selecting an accurate order, $P(\text{accurate})=\frac{996}{1134}$

Step3: Calculate probability of two accurate orders with replacement

Since the selections are made with replacement, the probability that both orders are accurate is the product of the probabilities of each order being accurate.
$P(\text{both accurate})=\frac{996}{1134}\times\frac{996}{1134}=\frac{996^{2}}{1134^{2}}=\frac{992016}{1285956}\approx0.7714$
Two events are independent if the occurrence of one does not affect the probability of the occurrence of the other. When sampling with replacement, the probability of getting an accurate order on the second draw is the same as the probability of getting an accurate order on the first draw. So the events are independent.

Answer:

The probability is $0.7714$. The events are independent.