Which of the following are binomial experiments? For those are not, indicate which part of the definition of binomial experiment does not apply.
Tossing a fair coin 1000 times and counting the number of times a head appears.
Tossing a fair coin and counting the number of tosses before a head appears.
Checking five students for drug use from a class of thirty students in which ten percent use drugs.
The correct answer and explanation is :
Answer:
- Tossing a fair coin 1000 times and counting the number of times a head appears
- This is a binomial experiment because it satisfies all four conditions of a binomial experiment:
- Fixed number of trials (1000 tosses).
- Two possible outcomes per trial (Head or Tail).
- The probability of success (getting heads) is constant at 0.5.
- The trials are independent.
- Tossing a fair coin and counting the number of tosses before a head appears
- This is not a binomial experiment because it does not have a fixed number of trials. Instead, the number of trials is random, stopping only when a head appears. This is actually a geometric experiment rather than a binomial one.
- Checking five students for drug use from a class of thirty students in which ten percent use drugs
- This is not strictly a binomial experiment because the probability of selecting a drug user changes after each selection (since students are not replaced). This violates the assumption of independent trials with constant probability. However, if the sample size were much smaller compared to the population, we could approximate it as binomial.
Explanation:
A binomial experiment must satisfy four conditions:
- Fixed number of trials (n): The experiment must have a predetermined number of trials.
- Only two possible outcomes per trial: Each trial results in either a success or a failure.
- Constant probability of success (p): The probability of success must remain the same for each trial.
- Independence: The outcome of one trial must not affect the outcome of another.
The first experiment (1000 coin tosses) meets all conditions, making it binomial.
The second experiment (counting tosses until first head) lacks a fixed number of trials, making it geometric instead.
The third experiment (student drug testing) lacks independence because the probability of selecting a drug user changes after each selection, making it a hypergeometric experiment rather than binomial.
