Question abount independence events and conditional events

In summary, the conversation discusses proving implications using the ration ideal in an intuitive way. It also raises a question about determining the probability of someone lying based on statements from others. The solution involves using the concept of independence and conditional probabilities to calculate the probability of C lying.
  • #1
Alexsandro
51
0
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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  • #2
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 ?
 
  • #3
[tex]A \perp B \longrightarrow P(A|B) = P(A) \longrightarrow \frac{P(A \& B)}{P(B)} = P(A) \longrightarrow[/tex]

[tex]\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 [/tex]

[tex]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[/tex]
 
  • #4
{p^2+(1-p)^2}/{3p^2+(1-p)^2}
 
  • #5
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"
 

What is the difference between independent events and conditional events?

Independent events are events that have no influence on each other. This means that the occurrence of one event does not affect the probability of the other event happening. Conditional events, on the other hand, are events that are influenced by each other. The probability of one event occurring is dependent on the occurrence of another event.

How do you calculate the probability of independent events?

To calculate the probability of independent events, you can use the multiplication rule. This rule states that the probability of two independent events occurring together is equal to the product of their individual probabilities. So, if the probability of event A is 0.5 and the probability of event B is 0.3, then the probability of both events occurring together is 0.5 x 0.3 = 0.15.

What is the difference between mutually exclusive events and independent events?

Mutually exclusive events are events that cannot occur at the same time. This means that if one event happens, the other event cannot happen. On the other hand, independent events can occur at the same time, and the occurrence of one event does not affect the other event's probability.

What is the definition of conditional probability?

Conditional probability is the probability of an event occurring, given that another event has already occurred. It is calculated by dividing the probability of the two events occurring together by the probability of the first event occurring.

How do you determine if two events are independent or not?

To determine if two events are independent, you can use the multiplication rule. If the probability of both events occurring together is equal to the product of their individual probabilities, then the events are independent. If not, then the events are dependent on each other.

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