2. A test has 20 multiple-choice questions, each with four options (A, B, C, D).
ID: 3255426 • Letter: 2
Question
2. A test has 20 multiple-choice questions, each with four options (A, B, C, D). Students guess at random for each question. Suppose after selecting an answer, the instructor randomly eliminates of the incorrect answers that the student did not pick. two answer the same or Students have the option to either (1) keep their (2) switch their answer to the only other remaining option. Student B Student Instructor randomly B randomly keeps answer guesses (c eliminates two incorrect OR Student D options switches answer a. What is the probability that a student gets the first question correct if the student keeps their answer the same? [1 mark] b. What is the probability that a student gets the first question correct if the student switches their answer? (Hint: assume that the correct answer is "A". How many scenarios result in a correct response if you switch?) [1 mark]Explanation / Answer
Answer to part a)
The probability of answering the first question correct without switching is when the the answer c is correct
inititally the probability of correct answer is 0.25
thus probability of incorrect answer is 1-0.25 = 0.75
.
Later when two options are eliminated the probability of correct answer becomes = 0.5
and the probability of incorrect answer becomes = 0.5
.
Thus if the student does not swtich and still get the correct answer this implies he initially chose the correct answer
Thus final probability = 0.25*0.5 = 0.125
.
Answer to part b)
In this case we need to find out if the student switches his answer then he gets the correct answer this means initially he chose a wrong answer
initially the probability of incorrect answer is 0.75
later he selects the correct answer the probability of selecting the correct answer later is 0.5
Thus total probability = 0.75 *0.5 = 0.375
.
Answer to part c)
We find that part b got higher probability 0.375. Thus the child must prefer to swtich
Accordingly the expected number of correct answers = n * p
We got n = 20
p = 0.375
.
On plugging the values we get:
Expected value = 20 * 0.375 = 7.5 or 8 questions
Thus expected number of correct answers is 8 out of 20
.
Answer to part d)
if 60% of the answers are correct
this means 20 *60% = 12 questions are correct
n = 20
x = 12
p = 0.125 [ probability of keeping the answer same]
Using the binomial probability we get:
P(X =12) = nCx * p^x * (1-p)^(n-x)
P(X =12) = 20C12 * (0.125)^12 * (0.875)^8
P(X=12) = 0.00000062987.
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