Questions: Choose all of the following experiments that are binomial experiments or can be reduced to a binomial experiment. Testing one brand of aspirin by using 10 people to determine whether it is effective Testing four different brands of aspirin to see which brands are effective Checking 1000 applicants to see whether they were admitted to White Oak College

Choose all of the following experiments that are binomial experiments or can be reduced to a binomial experiment. Testing one brand of aspirin by using 10 people to determine whether it is effective Testing four different brands of aspirin to see which brands are effective Checking 1000 applicants to see whether they were admitted to White Oak College
Transcript text: Choose all of the following experiments that are binomial experiments or can be reduced to a binomial experiment. Testing one brand of aspirin by using 10 people to determine whether it is effective Testing four different brands of aspirin to see which brands are effective Checking 1000 applicants to see whether they were admitted to White Oak College
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Solution

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Solution Steps

Step 1: Understanding Binomial Experiments

A binomial experiment must satisfy the following conditions:

  1. There are a fixed number of trials, \( n \).
  2. Each trial has only two possible outcomes: success or failure.
  3. The probability of success, \( p \), is the same for each trial.
  4. The trials are independent.
Step 2: Analyzing the First Experiment

Experiment: Testing one brand of aspirin by using 10 people to determine whether it is effective.

  • Fixed number of trials: Yes, \( n = 10 \).
  • Two outcomes: Yes, each person either finds the aspirin effective (success) or not (failure).
  • Constant probability: Yes, assuming the probability of effectiveness is the same for each person.
  • Independence: Yes, assuming the effectiveness for one person does not affect another.

This experiment satisfies all the conditions of a binomial experiment.

Step 3: Analyzing the Second Experiment

Experiment: Testing four different brands of aspirin to see which brands are effective.

  • Fixed number of trials: Yes, \( n = 4 \) (one trial for each brand).
  • Two outcomes: Yes, each brand is either effective (success) or not (failure).
  • Constant probability: No, the probability of effectiveness may differ for each brand.
  • Independence: No, the effectiveness of one brand may influence the perception of another.

This experiment does not satisfy the conditions of a binomial experiment because the probability of success is not constant across trials.

Step 4: Analyzing the Third Experiment

Experiment: Checking 1000 applicants to see whether they were admitted to White Oak College.

  • Fixed number of trials: Yes, \( n = 1000 \).
  • Two outcomes: Yes, each applicant is either admitted (success) or not (failure).
  • Constant probability: Yes, assuming the probability of admission is the same for each applicant.
  • Independence: Yes, assuming the admission of one applicant does not affect another.

This experiment satisfies all the conditions of a binomial experiment.

Final Answer

The experiments that are binomial experiments or can be reduced to a binomial experiment are:

  1. Testing one brand of aspirin by using 10 people to determine whether it is effective.
  2. Checking 1000 applicants to see whether they were admitted to White Oak College.

\\(\boxed{\text{Experiments 1 and 3 are binomial experiments.}}\\)

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