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	33	59	37	13

if two orders are selected, find the probability that they are both from restaurant d.
a. assume that the selections are made with replacement. are the events independent?
b. assume that the selections are made without replacement. are the events independent?
a. assume that the selections are made with replacement. are the events independent?
the probability of getting two orders from restaurant d is 0.0204. the events are independent because choosing the first order does not affect the probability of the choice of the second order. (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 of getting two orders from restaurant d is . the events independent because choosing the first order the probability of the choice of the second order. (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 in all restaurants.
\[

$$\begin{align*} \text{Total orders}&=(314 + 274+250 + 148)+(33 + 59+37 + 13)\\ &=(986)+(142)\\ &=1128 \end{align*}$$

\]

Step2: Calculate number of orders from Restaurant D

The number of orders from Restaurant D is \(148 + 13=161\).

Step3: Calculate probability without replacement

The probability that the first - order is from Restaurant D is \(P_1=\frac{161}{1128}\).
Since the first order is not replaced, the probability that the second order is from Restaurant D is \(P_2=\frac{160}{1127}\).
The probability that both orders are from Restaurant D is \(P = P_1\times P_2=\frac{161}{1128}\times\frac{160}{1127}=\frac{25760}{1261256}\approx0.0204\)
The events are not independent because choosing the first order affects the probability of the choice of the second order.

Answer:

The probability of getting two orders from Restaurant D is \(0.0204\). The events are not independent because choosing the first order affects the probability of the choice of the second order.