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

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