For a certain type of hay fever, Medicine H has a 30% probability of working.
In which distributions does the variable X have a binomial distribution?
Select each correct answer.
When the medicine is tried with six patients, X is the number of patients for whom the medicine does not work. **
When the medicine is tried with two patients, X is the number of patients for whom the medicine worked.
When the medicine is tried with six patients, X is the number of patients for whom the medicine worked. **
When the medicine is tried with two patients, X is the number of doses each patient needs to take.
** is my answer choices
To determine which distributions the variable X has a binomial distribution, we need to consider the characteristics of a binomial distribution:
1. There are a fixed number of trials, denoted by n.
2. Each trial has only two possible outcomes: success (denoted as S) or failure (denoted as F).
3. The probability of success, denoted as p, remains constant from trial to trial.
4. The trials are independent of each other.
Based on these characteristics, we can analyze each statement:
1. "When the medicine is tried with six patients, X is the number of patients for whom the medicine does not work."
In this scenario, the variable X represents the number of failures (patients for whom the medicine does not work). This matches the definition of a binomial distribution, with n = 6 trials and a fixed probability of failure (1 - p). So, this statement represents a binomial distribution.
2. "When the medicine is tried with two patients, X is the number of patients for whom the medicine worked."
In this case, the variable X represents the number of successes (patients for whom the medicine worked). Similarly, this matches the definition of a binomial distribution, with n = 2 trials and a fixed probability of success (p). So, this statement also represents a binomial distribution.
3. "When the medicine is tried with six patients, X is the number of patients for whom the medicine worked."
Similar to statement 2, this scenario considers the number of successes (patients for whom the medicine worked). The conditions are the same, with n = 6 trials and a fixed probability of success (p). Therefore, this statement also represents a binomial distribution.
4. "When the medicine is tried with two patients, X is the number of doses each patient needs to take."
This statement does not match the characteristics of a binomial distribution. It does not involve a fixed number of trials, nor does it represent a count of successes or failures.
So, the correct answers are:
- When the medicine is tried with six patients, X is the number of patients for whom the medicine does not work.
- When the medicine is tried with two patients, X is the number of patients for whom the medicine worked.
- When the medicine is tried with six patients, X is the number of patients for whom the medicine worked.