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Part A
Extend the table so that we have row and column totals as shown in the diagram below.
There are 986 accurate orders from all four restaurants combined, out of 1125 orders total. The probability of an accurate order is 986/1125. Selecting two such orders, with replacement, means we get a probability of
(986/1125)*(986/1125) = 0.7682 which is approximate.
Now to the question of independence vs dependence. It all comes down to this: Does the first selection affect the second selection? If so, then we have dependent events. Otherwise the events are independent. Because we put the first selection back (or replaced it with an equivalent order), this means the first selection does not alter the second selection. The two selections effectively exists in their own separate universe so to speak.
<h3>Answer: The probability is <u>0.7682</u> and the events <u>are</u> independent</h3>============================================================
Part B
This time, we aren't putting the order back in the pile. Whatever we select is taken out entirely without replacement. The first selection will affect the second one. The probability of getting an accurate order for the first selection is 986/1125 = 0.8764 while the second selection is slightly different at 985/1124 = 0.8763. This difference is enough to show that we have dependent events. The second event depends on the first.
Note the second fraction has each term drop by 1.
The probability of picking two accurate orders, when replacement is not made, is (986/1125)*(985/1124) = 0.7681 approximately
<h3>Answer: The probability is <u>0.7671</u> and the events are <u>not</u> independent.</h3>