Probability that A speaks truth is $\frac{4}{5}$ . A coin is tossed. A reports that a head appears. The probability that actually there was head is:
$\frac{4}{5}$
Let H denote the event that the coin shows a head and T that it shows a tail, with \(P(H)=P(T)=1/2\) for a fair coin.
\(P(\text{reports head}\mid H) = 4/5\) (A speaks the truth), and \(P(\text{reports head}\mid T) = 1-4/5 = 1/5\) (A lies).
By Bayes' theorem, \(P(H\mid\text{reports head}) = \dfrac{P(H)P(\text{reports head}\mid H)}{P(H)P(\text{reports head}\mid H)+P(T)P(\text{reports head}\mid T)}\).
\(= \dfrac{(1/2)(4/5)}{(1/2)(4/5)+(1/2)(1/5)} = \dfrac{2/5}{2/5+1/10} = \dfrac{2/5}{1/2} = \dfrac{4}{5}\).
Similar Questions
Bag I contains 3 Red and 4 Black Balls and Bag II contains 4 Red and 5 Black Balls. One ball is transferred from Bag I to Bag II and then a ball drawn from Bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black.
Important Questions from Baye's Theorem
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