Section 5.3 - Binomial Distributions

A probability distribution has the following characteristics:

1. It has parameters n (the number of trials) and p (the probability of success for a single trial).
2. Its mean is given by μ =n⋅p
3. Its variance is given by σ² = n⋅p(1-p) or μ⋅(1-p)
4. Its standard deviation is given by σ = √np(1-p)

     PROBLEM: A quiz consists of five multiple-choice questions, each with four possible answer choices (A, B, C, or D), one of which is correct. Suppose that an unprepared student does not read the question, but simply makes one random guess for each question. Let the random variable X equal the number of correct guesses the student makes for the five questions.

  a. Is the binomial distribution an appropriate model for the probability distribution of the random variable X? Yes or no? Justify your answer.

  b. Identify n and p.

  c. Calculate μ, σ², and σ

     SOLUTION: Yes, the binomial distribution is an appropriate model for the probability distribution of the given process because:

A

1. there are n identical trials: five random guesses;
2. each trial (guess) results in only two outcomes: correct or incorrect, where the outcome correct will denote a success;
3. p, the probability of success (correct) on a single trial (guess) is one out of four or 0.25 and is the same from trial to trial;
4. the trials (guesses) are independent since the outcome of one guess does not affect the outcome of any other;
5. the variable of interest is the number of successes (correct guesses) in the five trials.

B

n = 5 and p = 0.25

C

μ = n⋅p = 5⋅0.25 = 1.25

σ² = n⋅p(1-p) = 5⋅0.25(1-0.25) = 1.25(0.75) = 0.9375

σ = √0.9375 = 0.9682

Course Description

Journal