# probability exam question

Hi,

I know the answer to this question but I am not sure how they worked it out. I know the method to work it out.
'
Steve, Katerina and Jess are three students who have agreed to take part in a psychology experiment. Each student is to answer several sets of multiple-choice questions. Each set has the same number of questions, n, where n is a number greater than 20. For each question there are four possible options (A, B, C or D), of which only one is correct.

If Katerina answers a question correctly, the probability that she will answer the next question correctly is 3/4 . If she answers a question incorrectly, the probability that she will answer the next question incorrectly is 2/3
.
In a particular set, Katerina answers Question 1 incorrectly.
b. i. Calculate the probability that Katerina will answer Questions 3, 4 and 5 correctly.
four decimal places.
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Commented:
About i: you know the first question was answered incorrectly, so you have 2/3 chance that the second is answered correctly. Let's consider the two options separately:

1. Question 2 is answered correctly. The chance for that is 2/3. The chance to answer q3 correctly are now 3/4 (because q2 was correctly answered), then the chance for q4 is again 3/4 and for q5 it's again 3/4. This makes 2/3 x 3/4 x 3/4 x 3/4 = 0.28125

2. Question 2 is answered incorrectly. The chance for that is 1/3. The chance to answer q3 correctly are now 2/3 (because q2 was incorrectly answered), then the chances for q4 and q5 remain 3/4 (because the previous question was answered correctly). This makes 1/3 x 2/3 x 3/4 x 3/4 = 0.125

Now we sum both probabilities to get the total probability: 0.40625
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Commented:
if the probability of answering question n correctly is p,
then the probability of answering question n+1 correctly is
p*(3/4) + (1-p)*(1-2/3)  = (4+5*p)/12
the probability of answering question n+2 correctly would then be
(4+5*((4+5*p)/12))/12
etc.

after 25 questions the probability will be getting very close to satisfying
P  =  (4+5*P)/12
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Author Commented:
just to clarify, I didn't answer the question as the probability amounts were given.
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Commented:
"I know the answer to this question but I am not sure how they worked it out. I know the method to work it out."
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Author Commented:
The answer used a transition matrix and I am not sure how to get this
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Commented:
The transition matrix is likely an expanded version of http:#a39456288
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