Let's analyze the table you provided and the questions that follow.

The table gives the number of accurate and inaccurate orders for four different drive-thru restaurants (A, B, C, and D).

We are asked to find the probability that two randomly selected orders are both from Restaurant D, and answer two sub-questions:

1. For selections with replacement (and whether the events are independent).
2. For selections without replacement (and whether the events are independent).

Total Orders for Each Restaurant

• Restaurant D accurate orders: 141
• Restaurant D inaccurate orders: 15
• Total orders from Restaurant D = 141 + 15 = 156

The total number of orders across all restaurants is:

$315 + 35 + 276 + 52 + 230 + 36 + 141 + 15 = 1100$

Part a: With Replacement

If two orders are selected with replacement, the probability that both are from Restaurant D is:

$P(\text{both from D}) = P(\text{first from D}) \times P(\text{second from D}) = \left(\frac{156}{1100}\right) \times \left(\frac{156}{1100}\right)$

$P(\text{both from D}) = \left(\frac{156}{1100}\right)^2 = \frac{24336}{1210000} \approx 0.0201$

Since we are selecting with replacement, the two selections are independent events.

Part b: Without Replacement

If two orders are selected without replacement, the probability that both are from Restaurant D is:

$P(\text{first from D}) = \frac{156}{1100}$ $P(\text{second from D}) = \frac{155}{1099}$

Thus, the probability that both orders are from D is:

$P(\text{both from D}) = \frac{156}{1100} \times \frac{155}{1099} = \frac{156 \times 155}{1100 \times 1099} = \frac{24180}{1208900} \approx 0.0200$

Since we are selecting without replacement, the two selections are not independent, because the second probability changes based on the outcome of the first selection.

Let me know if you'd like further clarification or have more questions! Here are five related questions for further exploration:

1. What is the probability of selecting two orders from Restaurant A with replacement?
2. What is the probability of selecting one accurate order and one inaccurate order from Restaurant D with replacement?
3. How would the probability change if three orders were selected from Restaurant D with replacement?
4. Can you calculate the probability of selecting one order from Restaurant A and one from Restaurant B without replacement?
5. What would be the probability of selecting one order from Restaurant D and another from any other restaurant without replacement?

Tip:

When events are independent, the outcome of one event doesn't affect the probability of the next. When they are dependent, the outcome of one event influences the probability of subsequent events.