Charles is taking a multiple choice probability exam, and for each question, there are 3 possible answers, out of which only one is correct. Since the time is short, for any question, with probability 3/4 Charles decides to do the calculations and with probability 1/4 he just choses one of the 3 answers randomly. Whenever he decides to do the calculations, with probability 4/5 he will get the correct answer and with probability 1/5, he gets an answer which matches one of the 2 wrong answers. Suppose that he got a particular question wrong. What is the probability he actually did the calculations
+1223 there are two options: 1) he did the calculations and got it wrong or 2) he guessed and got it wrong.
1) 1/4 * 1/5 = 1/20
2) 3/4 * 2/3 = 1/2
The probability we want is the answer for #1 divided by the sum of #1 and #2.
\(\frac{1/20}{1/20 + 1/2} = \frac{0.05}{0.55} = \boxed{\frac{1}{11}}\)
