Answer
a. Rolling a die 10 times and observing the number of spots.
In this case, there is a total of 10 trials (rolling), and they are all identical. All 10 trials are performed under identical conditions. Here, n = 10. Furthermore, the trials are independent. The outcome of one trial does not affect the outcome of another trial.
However, each trial has more than two possible outcomes, which is 1,2,3,4,5 or 6. The probabilities of these six outcomes are not constant.
Since the second and third conditions of a binomial experiment are not satisfied, this is not an example of a binomial experiment.
b. Rolling a die 12 times and observing whether the number obtained is even or odd.
In this case, there are a total of 12 trials (rolling), and they are all identical. All 12 trials are performed under identical conditions. Here, n = 12. Each trial has two outcomes: odd number or even number. The probability of obtaining an odd number or an even number are the same, which is $\frac{1}{2}$. The sum of these probabilities is 1.0. What is more, the trials are independent. The result of any preceding trial has no bearing on the result of any succeeding trial.
Since this experiment satisfies all four conditions, it is a binomial experiment.
c. Selecting a few voters from a very large population of voters and observing whether or not each of them favors a certain proposition in an election when 54% of all voters are known to be in favor of this proposition.
This example consists of nonidentical trials. Each trial is performed with a number of voters that are not specific. Each trial has two outcomes: voter favors certain proposition in an election and voters are against the certain proposition in an election. 54% of all voters are known to be in favor of this proposition. Let p be the probability of a voter who favors this proposition and q be the probability that a voter does not favor this proposition. These two probabilities, p and q, do not remain constant for each selection of a voter because of the limited number (54%) of voters. The probability of each outcome changes with each selection depending on what happened in the previous selections.
Because p and q do not remain constant for each selection, the trials are not independent. The outcome of the first selection affects the outcome of the second selection.
Therefore, this is not an example of binomial experiment.
Work Step by Step
See above.
