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 $P(A_1 \cap B_1) = 0.075$, $P(A_1 \cap B_2) = 0.245$. Find $P(A_2 \cap B_3)$.
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 $P(A_2 \cap B3) = P(A_2)*P(B3)$ I get 0.24. Does this mean that the events are not independent? If so, how are these events not independent?
Thanks in advance.