Answer:
The value is [tex]P(A | W) = 0.941 [/tex]
Step-by-step explanation:
From the question we are told that
The probability that the student knows the answer to the question is [tex]P(A) = 0.8[/tex]
The probability that that the student will guess is [tex]P(G) = 0.2[/tex]
The probability that that the student get the correct answer given that the student guessed is [tex]P(W /G) = 0.25[/tex]
Here W denotes that the student gets the correct answer
Generally it a certain fact that if the student knows the answer he would get it correctly
So the probability the the student got answer given that he knows it is
[tex]P(W | A) = 1[/tex]
Generally from Bayes theorem we can mathematically evaluate the probability that the student knows the answer given that he got it correctly as follows
[tex]P(A | W) = \frac{ P(A) * P(W | A )}{ P(A) * P(W | A) + P(G) * P(W| G)}[/tex]
=> [tex]P(A | W) = \frac{ 0.8 * 1}{ 0.8 * 1+ 0.2 * 0.25}[/tex]
=> [tex]P(A | W) = 0.941 [/tex]
