Answers Probability

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This document illustrates the probabilities to randomly guess the correct answers in a multi choice questions test.

MCSA (Multiple Choice Single Answer)

A single MCSA question with a alternative answers, contains 1 correct answer and ( a - 1 ) wrong answers

  • the probability that a randomly selected s answer is the correct one is:
    P s = 1 = 1 a

  • the probability that a randomly selected s answer is the wrong one is:
    P s = 0 = 1 - P s = 1 = 1 - 1 a = a - 1 a

In a test composed of n independent MCSA questions containing a alternative answers each,

  • the probability to randomly select exactly q = k correct answers is:
    P q = k = n k P s = 1 k P s = 0 n - k = n k 1 a k a - 1 a n - k = n k a - 1 n - k a n
    0 k n

  • the probability to randomly select all the correct answers k = n is:
    P q = n = P s = 1 n = 1 a n

  • the probability to randomly select all the wrong answers k = 0 is:
    P q = 0 = P s = 0 n = a - 1 a n

  • the probability to randomly select more than x n correct answers is:
    P q > x n = k = x n n P q = k = k = x n n n k a - 1 n - k a n
    0 x 1

MCMA (Multiple Choice Multiple Answers) with partial score option

On a single MCMA question, when the "partial score for MCMA" option is set, each individual alternative answer can be considered an independent true/false question.

A MCMA test with n questions and a alternative answers for each question is equivalent to a test composed of n a MCSA questions with 2 alternative answers each.

  • the probability to randomly select exactly q = k correct answers is:
    P q = k = n a k 1 2 n a
    0 k n a

  • the probability to randomly select all the correct answers k = n a is equivalento to the probability to randomly select all the wrong answers k = 0 :
    P q = n a = P q = 0 = 1 2 n a

  • the probability to randomly select more than x n a correct answers is:
    P q > n a = k = x n a n a P q = k = k = x n a n a n a k 1 2 n a
    0 x 1

MCMA (Multiple Choice Multiple Answers) without partial score option

On a single MCMA question with a alternative answers, when the "partial score for MCMA" option is disabled, a question is considered correctly answered only when all the a alternative answers are set to the correct value.

  • the probability that a randomly selected s answer is the correct one is:
    P s = 1 = 1 2 a

  • the probability that a randomly selected s answer is the wrong one is:
    P s = 0 = 1 - P s = 1 = 1 - 1 2 a

In a test composed of n independent MCMA questions containing a alternative answers each,

  • the probability to randomly select exactly q = k correct answers is:
    P q = k = n k P s = 1 k P s = 0 n - k = n k 1 2 a k 1 - 1 2 a n - k
    0 k n

  • the probability to randomly select all the correct answers k = n is:
    P q = n = P s = 1 n = 1 2 a n

  • the probability to randomly select all the wrong answers k = 0 is:
    P q = 0 = P s = 0 n = 1 - 1 2 a n

  • the probability to randomly select more than x n correct answers is:
    P q > x n = k = x n n P q = k = n k 1 2 a k 1 - 1 2 a n - k
    0 x 1

Examples

Probabilities for a test composed of n = 15 questions containing a = 5 alternative answers each.

  all wrong > 1 2 correct > 2 3 correct > 3 4 correct > 4 5 correct all correct
MCSA 1 28.42 1 235.86 1 8,831.92 1 988,871.98 1 988,871.98 1 30,517,578,125
MCMA partial 1 37,778,931,862,957,161,709,568 1 2 1 382.54 1 276,084.59 1 12,596,052.18 1 37,778,931,862,957,161,709,568
MCMA non-partial 1 1.61 1 208,064,843.35 1 433,008,250,196.35 1 2,766,415,372,899,402.57 1 2,766,415,372,899,402.57 1 37,778,931,862,957,161,709,568
 

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