# Probability based on aggregated data with overlapping data points

Say I have 20 individuals:

• 2 of them did event $A$
• 8 others did event $B$
• 10 did both events $A$ and $B$

Therefore, I can build the following table:

  Did 1 action  Did 2 actions
A            2             10
B            8             10


Here is how I want to interpret this:

• Given that someone did event $A$, they have a 10/12 (80%) chance to have also done $B$
• Given that someone did event $B$, they have a 10/18 (55%) chance to have also done $A$
• On average, given that someone has done an event, $A$ or $B$, that person has a 67.5% chance to have done another

Now if I describe my data like this:

 Did 1 action  Did 2 actions
10             10


I'm not sure how to interpret this compared to the data above. In the first table, I think I have the probability of $A+B$ given $A$ and the probability of $A+B$ given $B$. In the other table, I have the probability of $A+B$ given ($A$ or $B$).

But if this is a real world example, and I want to know what is the likelihood that someone will commit the second action given that they have already committed one, what probability should I retain?

Can these figures be interpreted as probabilities at all or am I making a mistake there as well? Am I using the wrong metrics / methodology?

• Is $A+B$ the event $A \cup B$ [$A$ OR $B$ (or both)] or $A \cap B$ [$A$ AND $B$]? Commented Mar 28, 2017 at 15:58
• What I wrote as A+B meant A∩B, sorry for the confusion. I will correct this soon. Commented Mar 29, 2017 at 6:52