You are taking a multiple-choice quiz that consists of five questions. Each ques

Question

You are taking a multiple-choice quiz that consists of five questions. Each question has five possible answers, only one is correct. To complete the quiz, you randomly guess the answer to each question. Find the probability of guessing each of the following:

a. Exactly three answers correctly.
b. At least three answers correctly.
c. Less than three answers correctly.

True or False? Replacement is allowed in binomial experiments. Explain your answer.

Course Subject: Basic Concept of Statistics (MA120)

Please answer each question correctly, completely, with detail & on a college level....Thank you

Explanation / Answer

There are 5 options, so the probability of getting one right is p = 1/5 = 0.2.

a)

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    5      
p = the probability of a success =    0.2      
x = the number of successes =    3      
          
Thus, the probability is          
          
P (    3   ) =    0.0512 [ANSWER]

***********************

b)

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    5      
p = the probability of a success =    0.2      
x = our critical value of successes =    3      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   2   ) =    0.94208
          
Thus, the probability of at least   3   successes is  
          
P(at least   3   ) =    0.05792 [ANSWER]

***********************

c)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    5      
p = the probability of a success =    0.2      
x = our critical value of successes =    3      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   2   ) =    0.94208
          
Which is also          
          
P(fewer than   3   ) =    0.94208 [ANSWER]

************************************************

True or False? Replacement is allowed in binomial experiments. Explain your answer.

It is sometimes allowed. For example, when there are 5 balls in an urn, 2 are red and 3 are blue. You draw 9 balls with replacement. This is a binomial experiment.

However, it is sometimes not allowed, like in the exam example above, because then, the trials would not be independent anymore, if you keep answering the same questions over and over again.

Related Questions

Navigate

• Browse (All)
• Browse Y
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.