Questions: Determine if the following probability experiments represent binomial experiments. If not, explain why. If a probability experiment is a binomial experiment, state the number of trials, (n), and probability of success, (p). (a) According to the Chronicle of Higher Education, there is a (59 %) probability that a randomly selected incoming first-year student will graduate from college within 6 years. Suppose that 40 incoming first-year students are randomly selected. After 6 years, each student is asked whether he or she graduated. (b) An experiment is conducted in which a pair of dice are cast until their sum is 3. The number of throws required is recorded. (a) Select the correct choice below and, if necessary, fill in the answer boxes to complete your answer. A. Yes, because the experiment satisfies all the criteria for a binomial experiment, [ n= p= text and text integers or decimals rounded to two decimal text places as needed.) ] B. No, because the trials of the experiment are not independent because the probability of success differs from trial to trial. C. No, because the experiment is not performed a fixed number of times. D. No, because there are more than two mutually exclusive outcomes for each trial.

Determine if the following probability experiments represent binomial experiments. If not, explain why. If a probability experiment is a binomial experiment, state the number of trials, (n), and probability of success, (p). (a) According to the Chronicle of Higher Education, there is a (59 %) probability that a randomly selected incoming first-year student will graduate from college within 6 years. Suppose that 40 incoming first-year students are randomly selected. After 6 years, each student is asked whether he or she graduated. (b) An experiment is conducted in which a pair of dice are cast until their sum is 3. The number of throws required is recorded. (a) Select the correct choice below and, if necessary, fill in the answer boxes to complete your answer. A. Yes, because the experiment satisfies all the criteria for a binomial experiment, [ n= p= text and text integers or decimals rounded to two decimal text places as needed.) ] B. No, because the trials of the experiment are not independent because the probability of success differs from trial to trial. C. No, because the experiment is not performed a fixed number of times. D. No, because there are more than two mutually exclusive outcomes for each trial.
Transcript text: Determine if the following probability experiments represent binomial experiments. If not, explain why. If a probability experiment is a binomial experiment, state the number of trials, $n$, and probability of success, $p$. (a) According to the Chronicle of Higher Education, there is a $59 \%$ probability that a randomly selected incoming first-year student will graduate from college within 6 years. Suppose that 40 incoming first-year students are randomly selected. After 6 years, each student is asked whether he or she graduated. (b) An experiment is conducted in which a pair of dice are cast until their sum is 3 . The number of throws required is recorded. (a) Select the correct choice below and, if necessary, fill in the answer boxes to complete your answer. A. Yes, because the experiment satisfies all the criteria for a binomial experiment, \[ \begin{array}{l} n=\square \\ p=\square \text { and } \\ \text { integers or decimals rounded to two decimal } \\ \text { places as needed.) } \end{array} \] B. No, because the trials of the experiment are not independent because the probability of success differs from trial to trial. C. No, because the experiment is not performed a fixed number of times. D. No, because there are more than two mutually exclusive outcomes for each trial.
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Solution

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Solution Steps

Step 1: Analysis of Part (a)

The probability experiment involves selecting 40 incoming first-year students and determining whether each student graduates within 6 years. This experiment satisfies the criteria for a binomial experiment:

  1. Fixed number of trials: \( n = 40 \)
  2. Two possible outcomes: Each student either graduates (success) or does not graduate (failure).
  3. Constant probability of success: The probability of success \( p = 0.59 \) remains the same for each trial.
  4. Independent trials: The outcome of one student's graduation does not affect another's.

Using the formulas for a binomial distribution, we calculate the following:

  • Mean \( \mu = n \cdot p = 40 \cdot 0.59 = 23.6 \)
  • Variance \( \sigma^2 = n \cdot p \cdot q = 40 \cdot 0.59 \cdot (1 - 0.59) = 9.68 \)
  • Standard Deviation \( \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{40 \cdot 0.59 \cdot 0.41} \approx 3.11 \)
Step 2: Analysis of Part (b)

In this experiment, a pair of dice is cast until their sum equals 3. The number of throws required is recorded. This does not represent a binomial experiment because:

  • The number of trials is not fixed; it varies based on the outcome of each throw.

Final Answer

For part (a):

  • Number of trials \( n = 40 \)
  • Probability of success \( p = 0.59 \)
  • Mean \( \mu = 23.60 \)
  • Variance \( \sigma^2 = 9.68 \)
  • Standard Deviation \( \sigma = 3.11 \)

For part (b):

  • The answer is: No, because the experiment is not performed a fixed number of times.

Thus, the final answer is: \[ \boxed{A} \] for part (a) and \(\boxed{\text{No}}\) for part (b).

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