About $$ 80 \% $$ of babies born with a certain ailment recover fully. A hospital is caring for five babies born with this ailment. The random variable represents the number of babies that recover fully. Decide whether the experiment is a binomial experiment. If it is, identify a success, specify the values of $$ n $$, $$ p $$, and $$ q $$, and list the possible values of the random variable $$ x $$. Is the experiment a binomial experiment? Yes No

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Yes, the experiment is a **binomial experiment**. Here's the breakdown:

### **Criteria for a Binomial Experiment**
A binomial experiment must satisfy the following conditions:
1. **Fixed Number of Trials ($$n$$)**: There are a set number of trials.
2. **Two Possible Outcomes**: Each trial results in either a "success" or a "failure."
3. **Constant Probability ($$p$$)**: The probability of success remains the same for each trial.
4. **Independence**: The outcomes of the trials are independent of each other.

### **Application to the Given Scenario**
1. **Fixed Number of Trials ($$n$$)**:
   - The hospital is caring for **5 babies**.
   - Therefore, $$ n = 5 $$.

2. **Two Possible Outcomes**:
   - **Success**: A baby **recovers fully**.
   - **Failure**: A baby **does not recover fully**.

3. **Constant Probability ($$p$$)**:
   - The probability of a baby recovering fully is **80%**, so $$ p = 0.8 $$.
   - Consequently, the probability of not recovering is $$ q = 1 - p = 0.2 $$.

4. **Independence**:
   - Assuming that the recovery of one baby does not influence the recovery of another, the trials are **independent**.

### **Summary of Binomial Parameters**
- **Success Definition**: A baby recovers fully.
- **Number of Trials ($$n$$)**: 5
- **Probability of Success ($$p$$)**: 0.8
- **Probability of Failure ($$q$$)**: 0.2
- **Possible Values of the Random Variable ($$x$$)**: 
  $$
  x = 0, 1, 2, 3, 4, 5
  $$
  This represents the number of babies (out of five) that recover fully.

### **Conclusion**
All the criteria for a binomial experiment are met in this scenario. Therefore, it is indeed a **binomial experiment**.

**Answer:**  
**Yes**, the experiment is a binomial experiment.

- **Success:** A baby recovers fully.
- **Number of trials (n):** 5
- **Probability of success (p):** 0.8
- **Probability of failure (q):** 0.2
- **Possible values of $$ x $$:** 0, 1, 2, 3, 4, 5
Yes, the experiment is a binomial experiment. - **Success:** A baby recovers fully. - **Number of trials (n):** 5 - **Probability of success (p):** 0.8 - **Probability of failure (q):** 0.2 - **Possible values of $$ x $$:** 0, 1, 2, 3, 4, 5

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