5. Decide whether the following experiment is a binomial experiment. Explain your answer.
A researcher observes the gender of the next 100 babies born at a local hospital. The boys and girls are equally likely. The random variable represents the number of boys.
p boy = .5
p not boy = 1-p boy = .5
sure looks like a binomial experiment to me
lets find the probability of four boys out of ten births for example
P(10,4) = C(10,4) .5^4 * (1-.5)^6
= 10!/[4!(6!)] *(1/16)*(1/64)
= 210 / 1024
= .205
Well, that was fun :)
To determine if the given experiment is a binomial experiment, we need to check if it satisfies the four criteria for a binomial experiment.
1. The experiment consists of a fixed number of trials: Yes, the researcher observes the gender of the next 100 babies, which is a fixed number of trials.
2. Each trial has only two possible outcomes: Yes, each baby can either be a boy or a girl.
3. The outcomes of each trial are independent: Yes, the gender of each baby is independent of the gender of the other babies.
4. The probability of success (getting a boy) is the same for each trial: Yes, it is mentioned that boys and girls are equally likely.
Therefore, the given experiment satisfies all the criteria for a binomial experiment. The random variable represents the number of boys, making it a binomial random variable.
To determine whether the given experiment is a binomial experiment, we need to check if it meets the following criteria:
1. The experiment consists of a fixed number of identical trials: In this case, the researcher is observing the gender of the next 100 babies born at a local hospital. Since this number is fixed, the first criterion is met.
2. Each trial has two possible outcomes: The possible outcomes in this case are boy or girl, making it a binary choice. Therefore, the second criterion is met.
3. The probability of success (p) is the same for each trial: The question states that boys and girls are equally likely, meaning the probability of a boy or a girl is 0.5 or 50%. Therefore, the third criterion is met.
4. The trials are independent: Each baby's gender does not affect the gender of the next baby. Therefore, the fourth criterion is met.
Having confirmed that all four criteria are satisfied, we can conclude that the given experiment is a binomial experiment.