The binomial distribution

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Question from PY:

An on-line game called ‘Shop Quiz’ is held by an e-commerce platform, from Monday to Friday
every week. It consists of 8 multiple choice questions (MCQ) and each question has four options (A,
B, C, D). Only one option is the correct answer. People who are able to correctly answer all 8
questions are winners and will be awarded a number of on-line shopping credits.
Let X represent the number of questions that a person can answer correctly in a ‘Shop Quiz’.

1) Explain why Binomial distribution might NOT be a suitable distribution for the random variable
X.

Mr. Saul likes playing the quiz, however, he is afraid that he might not have the necessary knowledge
to answer the quiz questions. (The quiz questions cover a variety of topics including science, history,
entertainment, sports and geography, etc.) Therefore, he tries to win the game by simply guessing the
answers to each question.

2) Explain why X can be regarded as being binomially distributed in Mr. Saul’s case.

Hi,

For a Binomial distribution you have an experiment (trial) with two possible outcomes, sometimes called success and failure. Your experiment is answering a question and success is getting the correct answer. The experiment is repeated $n$ times independently (in your case $n = 8$). In each trial the probability of success is the same (usually called $p$). The random variable $X$ is the number of success in the $n$ trials.

Since Mr. Saul is guessing on each question and there are 4 possible answers with only one correct, on each trial the probability of success is $p = \large \frac{1}{4}.$ Hence in this case the random variable $X$ has a Binomial distribution. In the more general case described in 1) the probability of success might not be the same for each question. If the person taking the quiz is a sports fanatic then they will likely have a higher probability of getting a sports question correct than a history question.

For the remainder of the questions you sent I suggest you go to the Quandaries and Queries page and use the Quick Search to search for the term binomial distribution. You will see our responses to a number of questions we have received and hopefully these responses will help.

Penny

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