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.
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To determine the distributions in which the variable X has a binomial distribution, we need to identify the characteristics of a binomial distribution:
1. The number of trials is fixed.
2. Each trial is independent, meaning that the outcome of one trial does not affect the outcome of another.
3. There are only two possible outcomes for each trial, typically referred to as "success" and "failure."
4. The probability of success remains the same for each trial.
Let's analyze each scenario:
1. When the medicine is tried with six patients, and X is the number of patients for whom the medicine does not work:
In this case, the number of trials is fixed, each trial (patient) is independent, and there are two possible outcomes (the medicine works or does not work). Additionally, the probability of the medicine not working (failure) remains the same for each patient. Therefore, X has a binomial distribution.
2. When the medicine is tried with two patients, and X is the number of patients for whom the medicine worked:
Similar to the first scenario, we have a fixed number of trials, independent trials, and two possible outcomes (the medicine works or does not work). However, in this case, X represents the number of patients for whom the medicine worked (success), not the number for whom it did not work. Therefore, X does not have a binomial distribution.
3. When the medicine is tried with six patients, and X is the number of patients for whom the medicine worked:
This scenario meets all the criteria for a binomial distribution: fixed number of trials, independent trials, two possible outcomes, and a constant probability of success. Therefore, X has a binomial distribution.
4. When the medicine is tried with two patients, and X is the number of doses each patient needs to take:
This scenario does not fit the characteristics of a binomial distribution. The number of trials (doses) each patient needs to take is not fixed, as it can vary. Additionally, it is not a binary outcome but a count of doses. Therefore, X does not have a binomial distribution.
In summary, the scenarios with binomial distributions 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 six patients, X is the number of patients for whom the medicine worked.
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