Part 1: Probability of getting an order from Restaurant A or an order that is accurate

To solve this, we need to use the formula for the probability of the union of two events $A$ and $B$, which is:

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Where:

• $P(A)$ is the probability of selecting an order from Restaurant A.
• $P(B)$ is the probability of selecting an accurate order.
• $P(A \text{ and } B)$ is the probability of selecting an accurate order from Restaurant A (i.e., the intersection of the two events).

First, let's calculate the total number of orders:

$\text{Total Orders} = (324 + 273 + 232 + 148) + (37 + 50 + 32 + 18) = 977$

Step 1: Probability of selecting an order from Restaurant A, $P(A)$:

The number of orders from Restaurant A is:

$P(A) = \frac{\text{Orders from A}}{\text{Total Orders}} = \frac{324 + 37}{977} = \frac{361}{977} \approx 0.369$

Step 2: Probability of selecting an accurate order, $P(B)$:

The number of accurate orders is:

$P(B) = \frac{\text{Total Accurate Orders}}{\text{Total Orders}} = \frac{324 + 273 + 232 + 148}{977} = \frac{977}{977} = 0.977$

Step 3: Probability of selecting an accurate order from Restaurant A, $P(A \text{ and } B)$:

The number of accurate orders from Restaurant A is 324. So:

$P(A \text{ and } B) = \frac{324}{977} \approx 0.332$

Step 4: Calculate the probability of getting an order from Restaurant A or an accurate order:

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) = 0.369 + 0.977 - 0.332 = 1.014$

Since probabilities cannot exceed 1, this result indicates that the total probability is essentially 1, meaning that getting an order from Restaurant A or an accurate order is highly likely.

Part 2: Are the events disjoint?

Two events are disjoint if they cannot happen at the same time. In this case, it is possible to select an order from Restaurant A and for that order to be accurate (i.e., you can have both events occur simultaneously).

Thus, the events are not disjoint because it is possible to receive an accurate order from Restaurant A.

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Summary:

• The probability of getting an order from Restaurant A or an order that is accurate is approximately 1 (rounded to three decimal places).
• The events are not disjoint because it is possible to receive an accurate order from Restaurant A.

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Would you like further details or have any other questions? Here are some related questions for further exploration:

1. What is the probability of getting an order from Restaurant B or an accurate order?
2. How would the probability change if the total number of orders increased?
3. What is the probability of getting an inaccurate order from Restaurant C?
4. What is the combined probability of selecting an accurate order from Restaurants A and D?
5. If you randomly select two orders, what is the probability that both are accurate?

Tip: When dealing with "or" probabilities, remember to subtract the overlap (intersection) to avoid double-counting!