(10 pts.) Use the table below to answer the question that follows. Write your answers as simplified fractions.
Number of Steaks and Roasts in Plant
Cuts of Beef
With High Fat Content
With Low Fat Content
Total
Flank Steaks
67
433
500
Rump Roasts
203
347
550
Total
270
780
1050
A United States Department of Agriculture (USDA) inspector is grading cuts of beef at a meat packing plant.
If a piece of beef is selected at random, what is the probability that it will be:
(Round to four decimal places.)
a.) A Flank Steak with Low Fat Content?
b.) A Rump Roast or Flank Steak with High Fat Content?

We can use the given table to calculate the probabilities for each scenario. The table shows the breakdown of beef cuts by fat content:

| Cuts of Beef  | With High Fat Content | With Low Fat Content | Total |
|---------------|-----------------------|----------------------|-------|
| Flank Steaks  | 67                    | 433                  | 500   |
| Rump Roasts   | 203                   | 347                  | 550   |
| **Total**     | 270                   | 780                  | 1050  |

The total number of pieces of beef is \( 1050 \), so any probability will be calculated by dividing by this total.

### a.) **Probability of selecting a Flank Steak with Low Fat Content:**

To find this probability, we take the number of Flank Steaks with Low Fat Content and divide it by the total number of beef cuts:

\[
P(\text{Flank Steak with Low Fat Content}) = \frac{\text{Flank Steaks with Low Fat}}{\text{Total Pieces of Beef}} = \frac{433}{1050}
\]

Now, simplify the fraction:

\[
P(\text{Flank Steak with Low Fat Content}) = \frac{433}{1050} \approx 0.4124
\]

So, the probability is **0.4124**.

### b.) **Probability of selecting a Rump Roast or Flank Steak with High Fat Content:**

This requires adding the number of Rump Roasts and Flank Steaks with High Fat Content, then dividing by the total number of beef cuts:

\[
P(\text{Rump Roast or Flank Steak with High Fat Content}) = \frac{\text{Rump Roasts with High Fat} + \text{Flank Steaks with High Fat}}{\text{Total Pieces of Beef}} = \frac{203 + 67}{1050}
\]

Simplify the sum and the fraction:

\[
P(\text{Rump Roast or Flank Steak with High Fat Content}) = \frac{270}{1050} = \frac{9}{35} \approx 0.2571
\]

So, the probability is **0.2571**.

### Final Answers:
- **a)** The probability of selecting a Flank Steak with Low Fat Content is **0.4124**.
- **b)** The probability of selecting a Rump Roast or Flank Steak with High Fat Content is **0.2571**.

Would you like a deeper breakdown or have any questions?

### Related Questions:
1. What is the probability of selecting a Rump Roast with Low Fat Content?
2. What is the probability of selecting a cut of beef with High Fat Content?
3. What is the probability of selecting a Flank Steak (regardless of fat content)?
4. What is the probability of selecting any cut with Low Fat Content?
5. If two pieces of beef are selected randomly, what is the probability both are Flank Steaks?

### Tip:
When solving probability problems, always ensure the total number of outcomes is correct, especially when they are based on categories or conditional choices.

A United States Department of Agriculture (USDA) inspector is grading cuts of beef at a meat packing plant. If a piece of beef is selected at random, what is the probability that it will be: a.) A Flank Steak with Low Fat Content? b.) A Rump Roast or Flank Steak with High Fat Content?

Learn how to calculate the probability of selecting a Flank Steak with Low Fat Content or a Rump Roast with High Fat Content at a meat packing plant. This problem uses basic probability concepts, fractions, and ratios, suitable for students in Grades 6-8.

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