p(mom,upstairs)=0.6×0.7=0.42;
p(mom,downstairs)=0.6×0.3=0.18;
p(dad,upstairs)=0.4×0.50=0.20;
p(dad,downstairs)=0.4×0.50=0.20.
[Total probability=0.42+0.18+0.20+0.20=1 (all outcomes covered).]
(a) p(upstairs)=0.42+0.20=0.62;
(b) p(downstairs)=0.18+0.20=0.38 (both parents);
p(dad|downstairs)=0.20/0.38=0.53 approx.
[Total probability=0.62+0.38=1 (all outcomes covered).]
EXAMPLE
There are 100 presents.
Mom hides 60, dad hides 40.
Mom hides 42 (70% of 60) upstairs and 18 downstairs (30% of 60).
Dad hides 20 (50% of 40) upstairs and 20 downstairs (50% of 40).
There are 38 presents downstairs and 62 upstairs.
Out of the 38 downstairs, dad has hidden 20. This is a probability of 20/38=10/19.