Each flat tire that is brought into a tire shop is either fixed or replaced.  The table below lists the number of flat tires that were brought into​ Hollis's tire shop last month and where the tire was mounted when it went flat.  Use this inforation to answer the​ question(s) about​ Hollis's customers.
right front
left front
right rear
left rear
Total
fixed
870
781
511
568
2730
replaced
429
781
424
418
2052
Total
1299
1562
935
986
4782
 
Question content area bottom
Part 1
​(Round all answers to 5 decimal​ places)
A. If 8 tires are randomly selected with​ replacement, what is the probability that they are all replaced​?  
  
enter your response here
B. If 8 tires are randomly selected with​ replacement, what is the probability that they are all right rear tire​s? 

Let's solve both parts of the question step by step.

### Part A: Probability that all 8 tires selected are replaced

The total number of tires brought into the shop is \( 4782 \) (from the table).

The number of tires that were replaced is \( 2052 \).

The probability \( P(\text{replaced}) \) that a randomly selected tire is replaced is given by the ratio of replaced tires to total tires:

\[
P(\text{replaced}) = \frac{2052}{4782}
\]

Now, if we select 8 tires **with replacement**, the probability that all 8 are replaced is the product of the individual probabilities, i.e., \( P(\text{replaced})^8 \).

So,

\[
P(\text{all replaced}) = \left(\frac{2052}{4782}\right)^8
\]

Let's calculate this probability.

\[
P(\text{all replaced}) = \left(\frac{2052}{4782}\right)^8
\]

Now, I'll calculate this value.The probability that all 8 tires selected are replaced is approximately \( 0.00115 \) (rounded to 5 decimal places).

---

### Part B: Probability that all 8 tires selected are right rear tires

The number of right rear tires brought into the shop is \( 935 \) (from the table).

The probability \( P(\text{right rear}) \) that a randomly selected tire is a right rear tire is given by:

\[
P(\text{right rear}) = \frac{935}{4782}
\]

Again, if 8 tires are selected **with replacement**, the probability that all 8 are right rear tires is:

\[
P(\text{all right rear}) = \left(\frac{935}{4782}\right)^8
\]

Now, I'll calculate this value.It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, you can calculate the probability manually by following these steps:
1. The probability that a randomly selected tire is a right rear tire is \( P(\text{right rear}) = \frac{935}{4782} \).
2. To find the probability that all 8 selected tires are right rear tires, raise this probability to the power of 8: 
   \[
   P(\text{all right rear}) = \left(\frac{935}{4782}\right)^8
   \]

Let me know if you'd like further assistance, and I can also guide you through similar calculations.

---

### Follow-up Questions:
1. What is the probability that none of the 8 selected tires are replaced?
2. What is the probability that at least one of the selected tires is a left front tire?
3. If 4 tires are randomly selected, what is the probability that exactly 2 are fixed?
4. How does the probability change if the selections are made without replacement?
5. What is the probability that all 8 selected tires are either left or right front?

### Tip:
When working with probabilities, ensure that the total number of outcomes is clearly understood for accurate calculations.

Each flat tire that is brought into a tire shop is either fixed or replaced. The table below lists the number of flat tires that were brought into​ Hollis's tire shop last month and where the tire was mounted when it went flat. Use this information to answer the​ question(s) about​ Hollis's customers. 
right front
left front
right rear
left rear
Total
fixed
870
781
511
568
2730
replaced
429
781
424
418
2052
Total
1299
1562
935
986
4782

Part 1: 
A. If 8 tires are randomly selected with​ replacement, what is the probability that they are all replaced​?
B. If 8 tires are randomly selected with​ replacement, what is the probability that they are all right rear tire​s?

Learn how to calculate the probability that all 8 tires randomly selected are replaced or are right rear tires using basic probability rules. Explore the application of probability in real-world scenarios such as tire shops. Suitable for Grades 10-12.

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