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.
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

Total orders \(=339 + 280+233 + 131+33 + 50+40 + 11=1117\)

Step2: Probability with replacement (part a)

Probability of selecting an order from Restaurant D first time \(P(D_1)=\frac{131 + 11}{1117}=\frac{142}{1117}\)
Since with replacement, probability second time \(P(D_2)=P(D_1)=\frac{142}{1117}\)
By multiplication rule for independent events \(P(D_1\cap D_2)=P(D_1)\times P(D_2)=\frac{142}{1117}\times\frac{142}{1117}=\frac{142^{2}}{1117^{2}}\approx0.0162\)
Events are independent because replacing the first - selected order means the second - selection probability is not affected by the first.

Step3: Probability without replacement (part b)

\(P(D_1)=\frac{142}{1117}\)
After first selection (without replacement), total orders left \(n = 1117- 1=1116\)
If first order was from D, orders from D left \(m=142 - 1 = 141\)
\(P(D_2|D_1)=\frac{141}{1116}\)
By multiplication rule for dependent events \(P(D_1\cap D_2)=P(D_1)\times P(D_2|D_1)=\frac{142}{1117}\times\frac{141}{1116}=\frac{142\times141}{1117\times1116}\approx0.0161\)
Events are not independent because not replacing the first - selected order changes the probability of the second - selection.

Answer:

a. The probability is approximately \(0.0162\), and the events are independent.
b. The probability is approximately \(0.0161\), and the events are not independent.