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 up
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 possible outcomes (success or failure) and a constant probability of success. Among the given options, only rolling a six-sided number cube 24 times and recording if a 4 comes up satisfies these conditions. Therefore, the answer is rolling a six-sided number cube 24 times and recording if a 4 comes up. $\boxed{Rolling , a , six-sided , number , cube , 24 , times , and , recording , if , a , 4 , comes , up}
Explanation
Understand the properties of a binomial experiment A binomial experiment has the following properties:
The experiment consists of a fixed number of trials.
Each trial has only two possible outcomes: success or failure.
The probability of success is the same for each trial.
The trials are independent. We need to check which of the given options satisfies all these properties.
Analyze each option Let's analyze each option:
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 (rolling a 4): 6 1 (constant for each trial)
Trials are independent. This option satisfies all the properties of a binomial experiment.
Option 2: Rolling a six-sided number cube and recording the number until a 4 comes up.
The number of trials is not fixed. The experiment stops when a 4 comes up. This violates the first property. This is not a binomial experiment.
Option 3: Rolling a six-sided number cube and recording the number until an even number comes up.
The number of trials is not fixed. The experiment stops when an even number comes up. This violates the first property. This is not a binomial experiment.
Option 4: Rolling a six-sided number cube 24 times and recording the number that comes up.
Fixed number of trials: 24
However, there are six possible outcomes (1, 2, 3, 4, 5, 6), not two. This violates the second property. This is not a binomial experiment.
Conclusion Based on the analysis, only Option 1 satisfies all the properties of a binomial experiment.
Examples
Binomial experiments are useful in many real-world scenarios. For example, consider a quality control process where you inspect a fixed number of items from a production line and check if each item meets the required standards (success) or not (failure). The number of defective items in the sample follows a binomial distribution, which helps in assessing the overall quality of the production process. Another example is in medical research, where you administer a new drug to a group of patients and observe whether each patient experiences improvement (success) or not (failure).
The correct option that describes a binomial experiment is Option A: rolling a six-sided number cube 24 times and recording if a 4 comes up. This option satisfies all the necessary criteria, including a fixed number of trials, two possible outcomes, constant probability of success, and independence of trials. The other options either do not have a fixed number of trials or do not restrict outcomes to two scenarios.
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