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.
The variable X has a binomial distribution when the following conditions are met:
1. There are a fixed number of trials.
2. Each trial has only two possible outcomes, commonly referred to as success and failure.
3. The probability of success remains constant for each trial.
4. The trials are independent of each other.
Now let's consider each option:
Option 1: When the medicine is tried with six patients, X is the number of patients for whom the medicine does not work.
This scenario fulfills the conditions for a binomial distribution because:
- We have a fixed number of trials (six patients).
- Each trial has two outcomes (working or not working).
- The probability of success (medicine working) remains constant at 30%.
- The trials (patients) are assumed to be independent of each other.
Option 2: When the medicine is tried with two patients, X is the number of patients for whom the medicine worked.
This scenario also fulfills the conditions for a binomial distribution, for the same reasons stated above.
Option 3: When the medicine is tried with six patients, X is the number of patients for whom the medicine worked.
Again, this option satisfies all the conditions for a binomial distribution, as explained previously.
Option 4: When the medicine is tried with two patients, X is the number of doses each patient needs to take.
This scenario does not meet the criteria for a binomial distribution because the variable X represents the number of doses each patient needs to take, rather than the number of successful outcomes in a fixed number of trials with a constant probability of success.
In summary, the correct options that have a binomial distribution for variable X 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.