Probability (independent?) events Hi everyone I have a question about the following problem: Events for a family: A_1 = ski, A_2= does not ski, B_1 = has children but none in 8-16, B_2 = has some children in 8-16, and B_3 = has no children. Also, P(A_1)=0.4, P(B_2)=0.35, P(B_1)=0.25 and P(A_1∩B_1)=0.075, P(A_1∩B_2)=0.245. Find P(A_2∩B_3).

Freddy Friedman

Freddy Friedman

Answered question

2022-07-23

Probability (independent?) events
Hi everyone I have a question about the following problem:
Events for a family: A 1 = ski, A 2 == does not ski, B 1 = has children but none in 8-16, B 2 = has some children in 8-16, and B 3 = has no children. Also, P ( A 1 ) = 0.4, P ( B 2 ) = 0.35, P ( B 1 ) = 0.25 and P ( A 1 B 1 ) = 0.075, P ( A 1 B 2 ) = 0.245. Find P ( A 1 B 1 ) = 0.075.
Here is my solution:
Since P(A1 and B1) = 0.075, P(A2 and B1) = 0.25-0.075= 0.175. Also since P(A1 and B2) = 0.245, P(A2 and B2) = 0.35 - 0.245 = 0.105. From this we can find P(A2 and B3) which is 0.6-0.175-0.105 = 0.32. But when I use the formula for independent events formula P ( A 2 B 3 ) = P ( A 2 ) P ( B 3 ) I get 0.24. Does this mean that the events are not independent? If so, how are these events not independent?

Answer & Explanation

Sandra Randall

Sandra Randall

Beginner2022-07-24Added 17 answers

Yes, it means the events are not independent. If they were, you would could find a simpler solution, requiring less data. For instance, if you knew P ( A 2 ) and P ( B 3 ), you would already know the answer:
P ( A 2 B 3 ) = P ( A 2 ) P ( B 3 )
It is perfectly legal for events that appear not to have a direct connection to be independent. It just might be that you're looking at a small population (a small village?) where by pure chance you find a lot of families with small children who ski, but few families without children who ski. You could probably think up some "real world" rationalization for a correlation as well (like, people with children have less time and are less likely to ski, or - conversely - have more incentive to ski with their children).
By the way, since you got:
P ( A 2 B 3 ) > P ( A 2 ) P ( B 3 )
the two events are positively correlated. It means, intuitively, that if you select a random family, once you learn that A 2 holds then the probability of B 3 increases. Note that being positively correlated is symmetric, and correlation does not imply causation.
smuklica8i

smuklica8i

Beginner2022-07-25Added 3 answers

Thanks in advance.

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