Problem related to conditional probability

[Raghu121]

Question statement is in picture

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.

 

[Dr.Peterson]

Did you use the fact that 6 people were left in the building? That lets you calculate how many actual entries and exits there were.

I made a table with rows for event labels (IN, OUT) and columns for actual events (ENTRY, EXIT), with a row and a column for totals, and filled in the actual number in each cell. Then I could read off answers from that.

Your work doesn't make sense to me, to the extent that I understand it. For example, 90% is not the probability of (ENTER and IN), but of IN given ENTER.