Given that
P(Know)=.2 , P(Don’t know)=.8
P(Correct/Know)=.5 , P(Wrong/Know)=.5
P(Correct/ Don’t Know)=.25 , P(Wrong/ Don’t Know)=.75
We have to find P(Know/Correct)
By Bayes’ Theorem,
$P(Know/Correct)=\frac{P(Know \cap Correct )}{P(Correct)}$
$=\frac{P(Know)\times P(Correct/Know)}{P(Know\cap Correct)\dotplus P(Don't Know\cap Correct)}$
$=\frac{P(Know)\times P(Correct/Know)}{P(Know) \times P(Correct/Know)\dotplus P(Don't Know)\times P(Correct/(Don't Know)}$
$=\frac{.2 \times .5}{.2 \times .5 \dotplus .8 \times .25}$
$=\frac{.1}{.3}$
$=.333$