Husband and Wife Watching TV Problem (Probability) The Problem:
On the basis of past viewing records, an advertising executive has determined that during prime time, husbands are watching TV 40% of the time. It has also been determined that when the husband is watching TV, 60% of the time the wife is also watching. When the wife is not watching TV, 25% of the time the husband is watching TV. 
Let us define two events: event H = "husband is watching TV" and event W = "wife is watching TV".
(a) Express three given probabilities by using events H and W
(b) If the wife is watching TV, what is the probability that the husband is not watching?
(c) Are the events H and W independent? Explain your answer.

My Thoughts:
(a):


*

*P(W given H) = 0.6

*P(H) = 0.4

*P (H given 'not W') = 0.25


(b):


*

*P('not H' given W) = P('not H') x P(W given 'not H')...???


(c): I know that the three equivalent ways to check independence are:


*

*P(H given W) = P(H)

*P(W given H) = P(W)

*P(H and W) = P(H) and P(W)

 A: a) First of all in probability theory event $\{W$ given $H\}$ we write like $\{W|H\}$ and call it condition probability. So you're correct in the a) answers but not write it well. 
$$P\{H\}=0.4 $$
$$P\{W|H\}=0.6 $$
$$P\{H|W^c\}=0.25 $$
where by $W^c$ I denote complement of event $W$.
b) Here we must use Bayes theorem or Law of total probability in order to determine that probability with the information we have:
$$P\{H^c|W\}=\frac{P\{H^c,W\} }{P\{W\}}=\frac{0.12}{0.36}=\frac{1}{3}, $$
Then we calculate:
$$P\{W|H \}=0.6,$$
$$P\{W^c|H\}=1-P\{W|H \}=0.4 $$
$$P\{H \}=0.4$$
$$P\{H^c \}=1-P\{H\}=0.6 $$
$$P\{H,W\}=P\{W|H\}P\{H\}=0.24 $$
$$P\{H,W^c\}=P\{H\}-P\{H,W\}=0.16$$
$$0.16=P\{H,W^c\}=P\{W^c\}P\{H|W^c\}=P\{W^c\}\cdot 0.25 \implies P\{W^c\}=\frac{16}{25}=0.64$$
$$P\{W\}=1-0.64=0.36. $$
$$P\{H^c,W^c\}=P\{H^c|W^c\}P\{W^c\}=0.75\cdot 0.64=0.48$$
$$P\{H^c,W\}=P\{H^c\}-P\{H^c,W^c\}=0.12$$
c) They are of course dependent, because:
$$0.24=P\{H, W\}\neq P\{H\}P\{W\}=0.4\cdot0.36=0.144$$
Maybe I missed some number or something like that, but the idea is still standing. Good luck!
A: Hint:  Draw a Venn diagram.  Put in each region the percentage of the time it represents.  What percent of the total time are they both watching?  What percent is the husband watching alone?
