When alice spends the day with the babysitter, there is a 0.6 probability that she turns on the tv and watches a show. her little sister betty cannot turn the tv on by herself. but once the tv is on, betty watches with probability 0.8. tomorrow the girls spend the day with the babysitter?

The questions are;
a) What is the probability that both Alice and Betty watch TV tomorrow?
b) What is the probability that Betty watches TV tomorrow?
c) What is the probability that only Alice watches TV tomorrow?

the probability of alice turning on the tv is 0.6
the probability of betty watching the tv once the tv is on is 0.8
a) 
in probability when the word and is mentioned then that means that both the conditions should occur therefore the probability of both instances are multiplied
the probability of alice watching tv - 0.6
probability of betty watching tv is 0.6 * 0.8 = 0.48
since the probability of betty watching tv depends on both alice switching on the tv and of betty actually watching tv 
therefore the probability of both watching tv = 0.6 * 0.48 = 0.288

b)
probability of betty watching tv alone is dependent upon alice turning on the tv
this is conditional probability, where one condition is depedent on another condition, in this case both alice should turn on the tv and betty should watch tv.
Therefore we have to multiply the probabilities of both events
probability of betty watching tv as calculated above is 0.6 * 0.8 = 0.48

c) 
only alice watching tv means that betty doesn't watch
the probability of betty not watching the tv = 0.6 * 0.2 = 0.12
this too, 2 events should occur. Alice should switch on the tv and betty should not watch tv. Therefore these 2 probabilities should be multiplied
therefore probability of only alice watching tv = 0.6 * 0.12 = 0.072