About 

9090​% 

of babies born with a certain ailment recover fully. A hospital is caring for 

sixsix 

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.

To determine whether this situation is a binomial experiment, we need to check if it meets the criteria for a binomial experiment:

1. **Fixed number of trials**: The experiment consists of a fixed number of trials \( n \).
2. **Two possible outcomes**: Each trial results in one of two outcomes, usually called "success" or "failure."
3. **Constant probability of success**: The probability of success \( p \) is the same for each trial.
4. **Independence of trials**: The outcome of each trial does not affect the outcome of the other trials.

Let's analyze the problem based on these criteria:

- **Fixed number of trials**: The hospital is caring for six babies, so there are six trials. Thus, \( n = 6 \).
- **Two possible outcomes**: For each baby, the possible outcomes are "recover fully" (success) or "do not recover fully" (failure).
- **Constant probability of success**: The probability of a baby recovering fully is given as 90%, or \( p = 0.90 \). This probability is assumed to be the same for each baby.
- **Independence of trials**: Assuming the recovery of one baby does not affect the recovery of another, the trials are independent.

Since all four criteria are satisfied, this is indeed a **binomial experiment**.

### Identifying the Components of the Binomial Experiment

- **Success**: A "success" in this context is defined as a baby recovering fully from the ailment.
- **Number of trials (n)**: The number of babies, \( n = 6 \).
- **Probability of success (p)**: The probability that a baby recovers fully, \( p = 0.90 \).
- **Probability of failure (q)**: The probability that a baby does not recover fully, \( q = 1 - p = 1 - 0.90 = 0.10 \).

### Possible Values of the Random Variable \( x \)

The random variable \( x \) represents the number of babies that recover fully. Since \( x \) counts the number of successes in \( n = 6 \) trials, the possible values for \( x \) range from 0 to 6. 

Thus, the possible values of \( x \) are:
\[ x = 0, 1, 2, 3, 4, 5, 6 \]

### Summary
- **This is a binomial experiment**.
- **Success**: A baby recovers fully.
- **\( n = 6 \)**: Number of babies.
- **\( p = 0.90 \)**: Probability of a baby recovering fully.
- **\( q = 0.10 \)**: Probability of a baby not recovering fully.
- **Possible values of \( x \)**: \( 0, 1, 2, 3, 4, 5, 6 \).

Would you like more details on any part of this explanation, or do you have any specific questions?

### Related Questions to Explore
1. What is the expected number of babies that will recover fully?
2. How do you calculate the variance and standard deviation for this binomial distribution?
3. What is the probability that exactly 4 out of the 6 babies will recover fully?
4. What is the cumulative probability that at most 5 babies recover fully?
5. How does the binomial distribution change if the probability of recovery decreases to 80%?

### Tip
When dealing with binomial experiments, always ensure that the four key criteria (fixed number of trials, two outcomes, constant probability, and independence) are met to correctly identify and apply the binomial model.

Explore the concept of binomial experiments using a scenario involving babies recovering fully from a certain ailment. Learn about the criteria for binomial experiments, including fixed number of trials, two possible outcomes, constant probability of success, and independence of trials. Calculate probabilities and identify key variables such as n, p, q, and x. Suitable for advanced high school students and those interested in probability theory.

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