About 50​% of babies born with a certain ailment recover fully. A hospital is caring for seven 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 is a binomial experiment, we need to check if it meets the following criteria:

1. **Fixed number of trials (n):** The experiment involves a fixed number of babies, which is 7 in this case.
2. **Two possible outcomes per trial:** Each baby can either recover fully (success) or not (failure).
3. **Constant probability of success (p):** The probability of a baby recovering fully is 50% (or 0.5) for each baby.
4. **Independence of trials:** The recovery of each baby is assumed to be independent of the others.

Given that all these conditions are satisfied, this is indeed a binomial experiment.

### Identification of Components:
- **Success:** A baby recovering fully from the ailment.
- **Number of trials (n):** \( n = 7 \)
- **Probability of success (p):** \( p = 0.5 \)
- **Probability of failure (q):** \( q = 1 - p = 1 - 0.5 = 0.5 \)

### Possible Values of the Random Variable \( x \):
The random variable \( x \) represents the number of babies that recover fully. Since there are 7 babies, \( x \) can take any integer value from 0 to 7. Thus, the possible values of \( x \) are:
\[ x = 0, 1, 2, 3, 4, 5, 6, 7 \]

Would you like to go into more detail or have any other questions?

### 5 Related Questions:
1. How do you calculate the mean and standard deviation of a binomial distribution?
2. What is the binomial probability formula, and how can it be applied to this scenario?
3. Can you calculate the probability that exactly 5 out of the 7 babies recover?
4. What is the cumulative probability of at least 3 babies recovering fully?
5. How would the binomial experiment change if the probability of recovery were not constant?

**Tip:** In a binomial experiment, the sum of the probabilities of all possible outcomes always equals 1. This can be a helpful check when calculating probabilities.

Explore the concept of binomial experiments in probability theory with a focus on the recovery of babies born with a certain ailment. Learn about the binomial probability formula, the conditions for a binomial experiment, and calculate probabilities for different outcomes. Suitable for Grades 9-12.

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