\\\\(\ extbf{Step1: Find the probability that it is actually a six }\\\\) \\\\(\ ext{Given:-}\\\\) \\\\(\ ext{Speaks the truth \\\\)3\\\\( out of \\\\)4\\\\( times}\\\\) \\\\(\ ext{Throws a die and reports that it is six}\\\\) \\\\(\ ext{Probability that the man speaks the truth is}\\\\) \\\\( = P(A) =\\dfrac{3}{4}\\\\) \\\\(\ ext{The probability that the man lies is = }\\\\) \\\\(P(B) = 1-{P(A) = }\\bigg(1-\\dfrac{3}{4}\\bigg)=\\dfrac{1}{4}\\\\) \\\\(\ ext{probability of getting}\\\\) \\\\(6=\\dfrac{1}{6}\\\\) \\\\(\ ext{probability of not getting}\\\\) \\\\(6=\\dfrac{5}{6}\\\\) \\\\(P(A|B)=\\dfrac {P(B|A).P(A)}{P(B)}\\\\) \\\\(\ ext{where, }\\\\)\\\\(P(A|B)=\\\\) \\\\(\ ext{probability of A given B is true}\\\\) \\\\(P(B|A)=\\\\) \\\\(\ ext{probability of B given A is true}\\\\) \\\\(P(A), P(B)=\\\\) \\\\(\ ext{the independent probabilities of A and B}\\\\) \\\\(\ ext{Applying Bayes' theorem, we get the required probability }\\\\)\\\\(=\\dfrac{\\dfrac{1}{6}.\\dfrac{3}{4}}{\\dfrac{1}{6}.\\dfrac{3}{4}+\\dfrac{5}{6}.\\dfrac{1}{4}}\\\\) \\\\(=\\dfrac{3}{8}\\\\)\\\\(\ extbf{Hence, the probability that it is actually a six is }\\\\) \\\\(=\\dfrac{3}{8}\\\\)