Question
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
Ask by Frazier Owen. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
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
Solution
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
Reviewed and approved by the UpStudy tutoring team
Like
error msg
Explain 
Simplify this solution Extra Insights
Yes, this experiment is a binomial experiment! In a binomial experiment, we have fixed trials, two possible outcomes (success or failure), constant probability of success, and independent trials. Here, the success is defined as a baby recovering fully. For this scenario, \( n = 5 \) (the number of babies), \( p = 0.8 \) (the probability of recovery), and \( q = 1 - p = 0.2 \) (the probability of not recovering). The possible values of the random variable \( x \), which represents the number of babies that recover, can be \( 0, 1, 2, 3, 4, \) or \( 5 \).
