(Batman and Robin) The city of Gotham keeps propping up newer challenges for our dynamic duo. The criminals in the city are so organized that they have come up with a schedule to commit the crimes.
a. The Mad Hatter commits maddeningly despicable crimes from 10AM - 11AM
b. The Riddler does nasty things (that are hard to understand) as well from 11AM - 12PM
c. 12PM to 1PM is lunch hour - no crimes get committed then
d. The Scarecrow scares the living lights out of citizens of Gotham from 1PM - 2PM
e. The Bane makes a grand appearance from 2PM - 3PM
f. The Joker arrives fresh from his afternoon nap and wreaks havoc from 3PM - 4PM
g. Just as the evening is setting, The Catwoman commits thefts from 4PM - 5PM
To keep up with this busy schedule, Batman and Robin (separately and) independently choose a random shift and they go and defeat whichever criminal is operating in that shift. Note that each chooses just one shift. If they happen to choose the lunch hour then they don’t fight anyone and just go home - its a holiday for them!
1) What is the probability that at least one of Batman and Robin gets a holiday?
2) Bane is too powerful an adversary - to defeat him, Batman and Robin need to fight together. All other criminals can be defeated by Batman or Robin alone. What is the probability that Bane gets defeated?
3) Batman has a crush on Catwoman and would like to (ahem ahem) fight her alone. What is the probability of that?
4) In the above scenario, what is the probability that two criminals get defeated?