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	53	31	18

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 0.7714. 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 . 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 accurate and non - accurate orders. The total number of accurate orders is $332 + 268+250 + 146=1096$, and the total number of non - accurate orders is $36 + 53+31 + 18 = 138$. So the total number of orders is $1096+138 = 1234$.

Step2: Calculate probability without replacement

The probability that the first order is accurate is $P_1=\frac{1096}{1234}$. Since the selection is without replacement, for the second order, there are now $1096 - 1=1095$ accurate orders left and $1234 - 1 = 1233$ total orders left. So the probability that the second order is accurate given the first is accurate is $P_2=\frac{1095}{1233}$. Then the probability that both orders are accurate is $P = \frac{1096}{1234}\times\frac{1095}{1233}=\frac{1096\times1095}{1234\times1233}=\frac{1199120}{1521522}\approx0.7879$.
The events are not independent when the selection is without replacement because the outcome of the first selection affects the probability of the second selection.

Answer:

The probability is $0.7879$. The events are not independent.