To determine whether the random variable \( X \) has a binomial distribution, we need to examine the characteristics that define a binomial distribution:
- Fixed Number of Trials: The experiment must have a fixed number of trials. Each trial is an independent observation.
- Two Possible Outcomes: Each trial must have two possible outcomes (success and failure).
- Constant Probability: The probability of success must be the same for each trial.
- Independent Trials: Each trial must be independent of the others.
In this scenario:
- We are choosing 20 students from a class of 70 students, which means we are not conducting a fixed number of independent trials. Rather, we are selecting a sample from a finite population without replacement.
- The outcome we are observing is whether each selected student missed the first exam (success) or not (failure). So, there are two possible outcomes.
- However, since we are selecting without replacement, the probability of success changes as we sample more students. For instance, if a student who missed the exam is selected first, the probability of selecting another student who missed the exam might change because we are no longer sampling from the original unchanged population.
Given these points, we can conclude:
The random variable does not have a binomial distribution because the trials are not independent (the selection is done without replacement), and the probability of success changes with each selection.
Thus, the random variable \( X \) does not exhibit the necessary properties of a binomial distribution.
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