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

	a	b	c	d
order not accurate	35	60	31	11

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 \\(\square\\). the events \\(\square\\) 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 accurate orders and total orders.
Accurate orders: \(336 + 275 + 236 + 122 = 969\)
Non - accurate orders: \(35+60 + 31+11=137\)
Total orders: \(969 + 137=1106\)

Step2: Probability for without replacement

When selecting without replacement, the probability that the first order is accurate is \(\frac{969}{1106}\). After selecting one accurate order, the number of accurate orders left is \(969 - 1=968\) and the total number of orders left is \(1106 - 1 = 1105\).
The probability that the second order is accurate given the first was accurate is \(\frac{968}{1105}\).
The probability that both are accurate is \(\frac{969}{1106}\times\frac{968}{1105}\)
\[

$$\begin{align*} \frac{969\times968}{1106\times1105}&=\frac{969\times968}{1106\times1105}\\ &=\frac{938992}{1222130}\\ &\approx0.7684 \end{align*}$$

\]
For independence: When sampling without replacement, the outcome of the first selection affects the probability of the second selection. So the events are not independent.

Answer:

The probability is \(0.7684\). The events are not independent.