About $$ 50 \% $$ 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 Is the experiment a binomial experiment? What is a success in this experiment? Baby doesn't recover This is not a binomial experiment. Baby recovers

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UpStudy AI · Accepted Answer

Yes, the experiment described is a **binomial experiment**. Let's break down why this is the case and address each part of your question in detail.

### **1. Is the experiment a binomial experiment?**

**Yes,** the experiment is a binomial experiment. It satisfies all the necessary conditions for a binomial experiment:

1. **Fixed Number of Trials (n):** There are **5 babies** being observed, which means there are 5 independent trials.
2. **Two Possible Outcomes:** For each baby, there are only two outcomes:
   - **Success:** The baby recovers fully.
   - **Failure:** The baby does not recover.
3. **Constant Probability of Success (p):** The probability of a baby recovering is constant at **50%** (or **0.5**) for each trial.
4. **Independence:** The recovery of one baby does not affect the recovery of another; each trial is independent.

Since all these conditions are met, the experiment is indeed a binomial experiment.

### **2. What is a success in this experiment?**

In the context of this experiment:

- **Success:** A **baby recovering fully** from the ailment.

> *Note:* While it's possible to define "failure" as a baby not recovering, in binomial experiments, "success" typically refers to the outcome of primary interest—in this case, recovery.

### **3. Specify the values of n, p, and q**

- **n (Number of Trials):**  
  $$ n = 5 $$  
  *(There are five babies being observed.)*

- **p (Probability of Success):**  
  $$ p = 0.5 $$  
  *(Each baby has a 50% chance of recovering.)*

- **q (Probability of Failure):**  
  $$ q = 1 - p = 0.5 $$  
  *(There is a 50% chance a baby does not recover.)*

### **4. List the possible values of the random variable**

The random variable $$ X $$ represents the **number of babies that recover fully** out of the five. Therefore, the possible values of $$ X $$ are:

$$
X = 0, 1, 2, 3, 4, 5
$$

This means:
- **0 babies recover**
- **1 baby recovers**
- **2 babies recover**
- **3 babies recover**
- **4 babies recover**
- **All 5 babies recover**

### **Summary**

- **Is the experiment a binomial experiment?**  
  **Yes**, it satisfies all the criteria for a binomial experiment.

- **What is a success in this experiment?**  
  **A baby recovers fully.**

- **Values:**  
  - $$ n = 5 $$  
  - $$ p = 0.5 $$  
  - $$ q = 0.5 $$

- **Possible values of the random variable $$ X $$:**  
  $$ X = 0, 1, 2, 3, 4, 5 $$

This framework allows you to apply binomial probability formulas to determine the likelihood of a certain number of babies recovering fully out of the five being cared for.
Yes, the experiment is a binomial experiment. A success is a baby recovering fully. The values are $$ n = 5 $$, $$ p = 0.5 $$, and $$ q = 0.5 $$. The possible values of the random variable $$ X $$ are 0, 1, 2, 3, 4, and 5.

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