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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

Answer :

pedrocarratore

Answer:

0.4737 or 47.37%

Step-by-step explanation:

When guessing, Charlie has 2/3 probability of getting the question wrong since there is only on correct answer.

The probability that Charlie actually did the calculations given that he got the question wrong is determined by the probability that he did the calculations and got it wrong divided by the probability of him getting an answer wrong.

The probability of doing the calculations and getting it wrong is:

[tex]P(C_W) = \frac{3}{4}*\frac{1}{5}=0.15[/tex]

The probability of getting an answer wrong is given by the probability of guessing and getting it wrong added to the probability of doing the calculations and getting it wrong:

[tex]P(W) = P(C_W)+ P(G_W)\\P(W) = \frac{3}{4}*\frac{1}{5}+ \frac{1}{4}*\frac{2}{3}\\P(W) = 0.316667[/tex]

Therefore, the probability that Charlie actually did the calculations given that he got the question wrong is:

[tex]P(C|W) = \frac{P(C_W)}{P(W)}=\frac{0.15}{0.316667} \\P(C|W) = 0.4737[/tex]

The probability that he actually did the calculations is 0.4737 or 47.37%.

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