## Binomial Distribution: Example #1

Objective: Compute probabilities and quantiles for a binomial random variable representing the number of correct guesses on a multiple-choice test and visualize these quantities using the Binomial Applet.

Problem Description: A multiple choice test has four possible answers to each of 16 questions. A student quesses the answer to each question, i.e., the probability of getting a correct answer on any given question is 0.25. The conditions of the binomial experiment are assumed to be met: n = 16 questions constitute the trials; each question results in one of two possible outcomes (correct or incorrect); the probability of being correct is 0.25 and is constant if no knowledge about the subject is assumed; the questions are answered independently if the student's answer to a question in no way influences his/her answer to another question.

The number of correct answers, X, is distributed as a binomial random variable with parameters n = 16 and p = 0.25. The Binomial Applet below display the probabilities for each of the 17 possible outcomes, i.e., for X = 0, 1, ..., 16, in a line chart.

The student can expect 4 (= 16 x 0.25) correct answers if all questions are answered by guessing. This is the mean (or expected value) of the binomial random variable when n = 16 and p = 0.25. The standard deviation is 1.732 = sqrt(3) = sqrt(16 x 0.25 x 0.75). Approximately two-thirds of the students who guess on the exam will fall within one standard deviation of the mean, i.e., will get 3, 4, or 5 questions correct.

A student desires to know the probabilities of getting various numbers of correct answers by chance. The worst possible case is to get all answers wrong. This probability is computed by selecting x = ? in the Prob popup menu. Type 0 in the resulting dialog and click OK. The probability, displayed to the right of the Prob menu, is 0.01, i.e., the student has a probability of 0.99 (= 1 - 0.01) of getting at least one question correct.

A student passes the exam if he/she gets at least 11 correct answers out of the 16. Select x >= a from the Prob popup menu and type 11 in the resulting dialog. Then P( x >= 11 ) = 2.0E-4 or, said another way, the student only has 2 chances in 10,000 of passing if he/she guesses.

The probability a student gets 3, 4, or 5 correct answers by guessing (i.e., the probability a student's number of correct answers is within one standard deviation of the mean) is computed by selecting a <= x <= b from the Prob popup menu. Enter 3 in the lower bound and 5 in the upper bound field. The resulting probability is 0.6132, which is not too far from the value of 2/3 mentioned above.

Click on F(x) to reveal the binomial cumulative distribution function for n = 16 and p = 0.25. The median number of correct answers (displayed by default) is 4, which is the same as the mean. Click on the q left arrow to get the lower quartile value of 3 correct answers and then click on the q right arrow to get the upper quartile of 5 correct answers.

The mean and the median are both 4, which indicates no skewness (according to our definition of skewness). However, the distribution is positively skewed as can be seen from the probability distribution plot. (Click on f(x) to return to the binomial probability plot.) The distribution of a binomial random variable is only symmetric when p = 1/2 for any given n Click on the p right arrow until p = 0.5 to show symmetry, which is the distribution of correct answers when a student guesses on a 16 question True/False test. In this case, P( 8 - c) = P( 8 + c) for c = 1, 2, ..., 8 and 8 is the mean of the distribution. For example, P( x = 6 ) = P( x = 10 ) = 0.1221.