To analyze the problem, let’s break it down step by step.

Given Data:

The table presents data on drive-thru order accuracy for four restaurants (A, B, C, D). We have the following:

Total orders for each restaurant:

• Restaurant A: $317 + 31 = 348$
• Restaurant B: $271 + 52 = 323$
• Restaurant C: $239 + 36 = 275$
• Restaurant D: $137 + 20 = 157$

Total number of orders across all restaurants: $348 + 323 + 275 + 157 = 1103$

Problem Breakdown:

We are asked to find the probability that two randomly selected orders are both from Restaurant D.

Part (a): With Replacement

When sampling with replacement, the probability for each order remains the same because after each selection, the total population does not change.

The probability that the first order is from Restaurant D is: $P(\text{first from D}) = \frac{157}{1103}$ Since we are sampling with replacement, the probability for the second order being from Restaurant D is also: $P(\text{second from D}) = \frac{157}{1103}$ Thus, the probability that both orders are from Restaurant D (independent events) is: $P(\text{both from D}) = \frac{157}{1103} \times \frac{157}{1103} = \left( \frac{157}{1103} \right)^2$

Part (b): Without Replacement

When sampling without replacement, after selecting the first order from Restaurant D, the total number of orders and the number of orders from Restaurant D decrease by 1.

• The probability that the first order is from Restaurant D is: $P(\text{first from D}) = \frac{157}{1103}$
• Once one order from Restaurant D has been selected, there are now 156 orders from Restaurant D left, and 1102 orders in total, so the probability that the second order is from Restaurant D is: $P(\text{second from D}) = \frac{156}{1102}$ Thus, the probability that both orders are from Restaurant D (dependent events) is: $P(\text{both from D}) = \frac{157}{1103} \times \frac{156}{1102}$

Independence of Events:

• With replacement: The events are independent, as the probability of selecting Restaurant D does not change after each selection.
• Without replacement: The events are not independent, since the outcome of the first selection affects the second selection (the total number of orders and orders from D changes).

Would you like more details or have any questions about the solution?

Related Questions:

1. What is the probability of selecting two accurate orders from Restaurant A with replacement?
2. What is the probability of selecting two inaccurate orders from Restaurant B without replacement?
3. How would the probabilities change if three orders were selected with replacement?
4. What is the total probability of selecting an accurate order from any restaurant?
5. How can conditional probability be applied to this scenario?

Tip:

When solving probability problems involving "without replacement," always adjust the total number of items after each selection to account for the change in population size.