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	36	58	37	19

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 0.7497. 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 \\(\boldsymbol{\
abla}\\) 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: \(320 + 274 + 246 + 128 = 968\)
Total orders: \((320 + 36)+(274 + 58)+(246 + 37)+(128 + 19)= 356+332+283+147 = 1118\)

Step2: Probability without replacement

For the first selection, probability of accurate order is \(\frac{968}{1118}\).
After first selection (without replacement), number of accurate orders left is \(968 - 1 = 967\), total orders left is \(1118 - 1 = 1117\).
So the probability that both are accurate is \(\frac{968}{1118}\times\frac{967}{1117}\).

Calculate \(\frac{968\times967}{1118\times1117}\):
\(968\times967 = 968\times(900 + 60 + 7)=968\times900+968\times60+968\times7 = 871200+58080+6776 = 936056\)
\(1118\times1117=(1100 + 18)\times(1100 + 17)=1100\times1100+1100\times17+18\times1100+18\times17 = 1210000+18700+19800+306 = 1248806\)
Then \(\frac{936056}{1248806}\approx0.7496\) (rounded to four decimal places)

For independence: When sampling without replacement, the outcome of the first selection affects the second (since total and accurate counts change), so events are not independent.

Answer:

The probability is \(0.7496\). The events are not independent.