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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.

\begin{tabular}{|l|c|c|c|c|}
\hline
& A & B & C & D \\
\hline
Order Accurate & 331 & 271 & 246 & 128 \\
\hline
Order Not Accurate & 31 & 51 & 33 & 13 \\
\hline
\end{tabular}

If one order is selected, find the probability of getting an order that is not accurate or is from Restaurant C.

Are the events of selecting an order that is not accurate and selecting an order from Restaurant C disjoint events?

The probability of getting an order from Restaurant C or an order that is not accurate is [tex]$\square$[/tex] .

(Round to three decimal places as needed.)

Sagot :

To solve this problem, we need to calculate a couple of probabilities using the given table data. Let's break down the solution step by step:

1. List the given values:

- Orders from Restaurant A: Accurate = 331, Not Accurate = 31
- Orders from Restaurant B: Accurate = 271, Not Accurate = 51
- Orders from Restaurant C: Accurate = 246, Not Accurate = 33
- Orders from Restaurant D: Accurate = 128, Not Accurate = 13

2. Calculate total orders for each restaurant:

- Total orders from Restaurant A: [tex]\(331 + 31 = 362\)[/tex]
- Total orders from Restaurant B: [tex]\(271 + 51 = 322\)[/tex]
- Total orders from Restaurant C: [tex]\(246 + 33 = 279\)[/tex]
- Total orders from Restaurant D: [tex]\(128 + 13 = 141\)[/tex]

3. Calculate total accurate and not accurate orders:

- Total accurate orders: [tex]\(331 + 271 + 246 + 128 = 976\)[/tex]
- Total not accurate orders: [tex]\(31 + 51 + 33 + 13 = 128\)[/tex]

4. Calculate overall total orders:

- Overall total orders: [tex]\(362 + 322 + 279 + 141 = 1104\)[/tex]

5. Find the probability of getting an order that is not accurate:

[tex]\[ \text{Probability of not accurate order} = \frac{\text{Total not accurate orders}}{\text{Overall total orders}} = \frac{128}{1104} \approx 0.116 \][/tex]

6. Find the probability of getting an order from Restaurant C:

[tex]\[ \text{Probability of order from Restaurant C} = \frac{\text{Total orders from Restaurant C}}{\text{Overall total orders}} = \frac{279}{1104} \approx 0.253 \][/tex]

7. Find the number of not accurate orders from Restaurant C:

- Not accurate orders from Restaurant C: [tex]\(33\)[/tex]

8. Find the probability of getting an order that is not accurate and from Restaurant C:

[tex]\[ \text{Probability of not accurate order from Restaurant C} = \frac{\text{Not accurate orders from Restaurant C}}{\text{Overall total orders}} = \frac{33}{1104} \approx 0.030 \][/tex]

9. Calculate the probability of getting an order that is not accurate or is from Restaurant C (using the inclusion-exclusion principle):

[tex]\[ \text{Probability of not accurate or Restaurant C} = \text{Probability of not accurate} + \text{Probability of Restaurant C} - \text{Probability of not accurate and Restaurant C} \][/tex]

[tex]\[ = 0.116 + 0.253 - 0.030 = 0.339 \][/tex]

10. Determine if the events are disjoint:

- Events are disjoint if they cannot happen at the same time. Since the probability of getting an order that is not accurate and from Restaurant C is not zero (0.030), these events are not disjoint.

So, the probability of getting an order that is not accurate or is from Restaurant C is [tex]\(0.339\)[/tex].

The events of selecting an order that is not accurate and selecting an order from Restaurant C are not disjoint.

[tex]\(\boxed{0.339}\)[/tex]