# Question abount independence events and conditional events

Alexsandro
Prove this questions using ration ideal in intuitive way.

Prove this implications and explain the results:

(a) A _|_ B => not A _|_ not B, onde _|_ means that events A and B are independent.

(b)[ P(A|C) >= P(B|C) ] and [ P(A|not C) >= P(B|not C) ] ==> P(A) > P(B)

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Alexsandro
Question about who tells the truth.

Interesting question:

" A says that B told him that C lied ".

If each of these person tells the truth with probability p, what is the probability that C lied ?

Homework Helper
$$A \perp B \longrightarrow P(A|B) = P(A) \longrightarrow \frac{P(A \& B)}{P(B)} = P(A) \longrightarrow$$

$$\frac{P(A) - P(A \& \~B)}{1 - P(\~B)} = P(A) \longrightarrow \frac{[1 - P(\~A)] - [P(\~B) - P(\~A \& \~B)]}{1 - P(\~B)} = 1 - P(\~A) \longrightarrow$$

$$1 - P(\~A) - P(\~B) + P(\~A \& \~B) = 1 - P(\~A) - P(\~B) + P(\~A)P(\~B) \longrightarrow \frac{P(\~A \& \~B)}{P(\~B)} = P(\~A) \longrightarrow \~A \perp \~B$$

balakrishnan_v
{p^2+(1-p)^2}/{3p^2+(1-p)^2}

balakrishnan_v
The above is the answer to the question
"Given " A says that B told him that C lied ".,what is the pr that c lied"