To determine whether the given random variables follow a binomial distribution, we need to consider the properties of a binomial distribution. A random variable \(X\) has a binomial distribution if it satisfies the following criteria:
- The number of trials \(n\) is fixed.
- Each trial has two possible outcomes, often termed "success" and "failure."
- The probability of success \(p\) is the same for each trial.
- The trials are independent.
Now, let's analyze each scenario:
A) Ten students are chosen from a statistics class (with replacement) of 400 students. Let \(X\) be the number of students who passed the class.
- This scenario involves selecting 10 students with replacement, which means each selection is an independent trial.
- There are two outcomes for each student: they either passed or did not pass.
- The probability of selecting a student who passed remains constant throughout the trials.
- Thus, this situation meets all the criteria for a binomial distribution.
B) A die is tossed three times. Let \(X\) be the sum of the three numbers obtained.
- Here, we are not considering the number of successes/failures; instead, we are looking at the sum of outcomes from three rolls of a die.
- The outcome for each trial is a number from 1 to 6, which does not meet the requirement of two possible outcomes (success/failure).
- Therefore, this situation does not represent a binomial distribution.
C) A coin is tossed until a head appears. Let \(X\) be the number of tosses.
- This scenario involves a variable number of trials since the tossing continues until a head is obtained.
- The number of trials is not fixed, violating the binomial distribution's criteria.
- Hence, this situation does not represent a binomial distribution either.
Based on this analysis, only scenario A follows a binomial distribution.
The correct answer is: only A is a binomial distribution.
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