Which of the following is a binomial experiment?
A. Rolling a six-sided number cube 24 times and recording if a 4 comes up
B. Rolling a six-sided number cube and recording the number until a 4 comes up
C. Rolling a six-sided number cube and recording the number until an even number comes
D. Rolling a six-sided number cube 24 times and recording the number that comes up
Answer (2)
A binomial experiment has a fixed number of independent trials, each with two outcomes (success or failure) and a constant probability of success.
Rolling a six-sided die 24 times and recording if a 4 comes up meets these criteria: fixed trials, two outcomes (4 or not 4), constant probability of success ( 6 1 ), and independent trials.
The other options do not meet all the criteria for a binomial experiment because they either don't have a fixed number of trials or don't have only two possible outcomes.
Therefore, the correct answer is: Rolling a six-sided number cube 24 times and recording if a 4 comes up. R o ll in g a s i x − s i d e d n u mb er c u b e 24 t im es an d recor d in g i f a 4 co m es u p
Explanation
Analyze the problem Let's analyze each option to determine which one fits the definition of a binomial experiment. A binomial experiment must have a fixed number of trials, each trial must have only two possible outcomes (success or failure), the probability of success must be the same for each trial, and the trials must be independent.
Analyze option 1 Option 1: Rolling a six-sided number cube 24 times and recording if a 4 comes up.
Fixed number of trials: 24
Two outcomes: Rolling a 4 (success) or not rolling a 4 (failure)
Probability of success: 6 1 for each trial
Trials are independent This fits the definition of a binomial experiment.
Analyze option 2 Option 2: Rolling a six-sided number cube and recording the number until a 4 comes up.
The number of trials is not fixed, as we continue rolling until a 4 comes up. This does not fit the definition of a binomial experiment.
Analyze option 3 Option 3: Rolling a six-sided number cube and recording the number until an even number comes.
The number of trials is not fixed, as we continue rolling until an even number comes. This does not fit the definition of a binomial experiment.
Analyze option 4 Option 4: Rolling a six-sided number cube 24 times and recording the number that comes up.
Fixed number of trials: 24
However, each trial has six possible outcomes (1, 2, 3, 4, 5, or 6), not two. This does not fit the definition of a binomial experiment.
Conclusion Therefore, the only option that is a binomial experiment is rolling a six-sided number cube 24 times and recording if a 4 comes up.
Examples
Binomial experiments are useful in many real-world scenarios. For example, consider a quality control process in a factory where items are randomly selected and inspected. Each item is either defective or not defective. If we inspect a fixed number of items and the probability of an item being defective is constant, then this is a binomial experiment. We can use the binomial distribution to calculate the probability of finding a certain number of defective items in the sample.
The correct answer is that rolling a six-sided number cube 24 times and recording if a 4 comes up is a binomial experiment. It meets all necessary criteria: fixed trials, two outcomes, constant probability, and independent trials. Other options do not satisfy these criteria.
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