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	38	52	34	15

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 . 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 by adding all values in the table.
$315 + 277+230 + 120+38+52+34+15=1081$

Step2: Calculate probability of selecting an order from Restaurant D on first draw (with - replacement)

The number of orders from Restaurant D is $120 + 15=135$. The probability of selecting an order from Restaurant D on the first draw, $P_1$, is $\frac{135}{1081}$.

Step3: Calculate probability of selecting an order from Restaurant D on second draw (with - replacement)

Since the selection is made with replacement, the probability of selecting an order from Restaurant D on the second draw, $P_2$, is also $\frac{135}{1081}$.

Step4: Calculate the probability of both events (with - replacement)

By the multiplication rule for independent events, the probability that both orders are from Restaurant D when selection is with replacement is $P = P_1\times P_2=\frac{135}{1081}\times\frac{135}{1081}=\frac{18225}{1168561}\approx0.0156$.
The events are independent because choosing the first order does not affect the probability of the choice of the second order.

Step5: Calculate probability of selecting an order from Restaurant D on first draw (without - replacement)

The probability of selecting an order from Restaurant D on the first draw, $P_1$, is $\frac{135}{1081}$.

Step6: Calculate probability of selecting an order from Restaurant D on second draw (without - replacement)

After the first order from Restaurant D is selected without replacement, there are $135 - 1 = 134$ orders from Restaurant D left and $1081-1 = 1080$ orders in total. So the probability of selecting an order from Restaurant D on the second draw, $P_2$, is $\frac{134}{1080}$.

Step7: Calculate the probability of both events (without - replacement)

By the multiplication rule for dependent events, the probability that both orders are from Restaurant D when selection is without replacement is $P=\frac{135}{1081}\times\frac{134}{1080}=\frac{18090}{1167480}\approx0.0155$.
The events are dependent because choosing the first order affects the probability of the choice of the second order.

Answer:

a. The probability of getting two orders from Restaurant D is $0.0156$. The events are independent because choosing the first order does not affect the probability of the choice of the second order.
b. The probability of getting two orders from Restaurant D is $0.0155$. The events are dependent because choosing the first order affects the probability of the choice of the second order.