# Problem related to Conditional probability

1. A detection event having occurred on Dec 16, 2019 before 7pm, is drawn at random. What is the probability that this event was triggered by a person having actually LEFT the building ?

2. A detection event having occurred on Dec 16, 2019 before 7pm, is drawn at random. If the sensor has labeled this event as IN. What is the probability that this event was triggered by a person having actually ENTERED the building ?

For the second part i tried to use conditional probability, considering 0.9 to be the probability of event being labeled IN & triggered by ENTRY. then the probability of IN = 0.9(IN & shows IN) + 0.06 (OUT & shows IN) =0.96. Thus final probability = 0.9/0.96 =0.9375 But when i tried the same approach for first part my answers were not coming out correct as i was getting probabilities >1. Would be great is someone can help me solve this.

• This is an interesting question! It seems phrased as a self-study or homework problem, so can you please add the self-study tag? Also, what have you tried to solve these questions so far? – Maurits M Aug 7 '20 at 11:20

## 1 Answer

Hints:

• There were $$100$$ events in total. How many more were actually ENTEREDs than LEFTs? So what was the probability of a detection actually being an ENTERED? What was the probability of a detection actually being a LEFT?

• What was the joint probability a detection was an IN triggered by an ENTERED? (It is not $$0.9$$ and is closer to $$0.5$$, as it has to be less than the probability of an ENTERED.) What was the the probability a detection was an IN triggered by a LEFT? (Those two add up to the probability of an IN.) So given that a detection was an IN, what was the conditional probability it was triggered by an actual ENTERED?