In a certain habitat, animals live in groups of two and groups of three. Each time a new animal
arrives at the habitat, it randomly chooses a group. If the chosen group has two animals in
it, the new animal joins the group forming a group of three animals. If the chosen group has
three animals in it, the new animal takes one of the animals away from the group of three and
forms a separate group of two animals.
If initially there are five animals in the habitat, and new animals arrive one at a time, what is
the probability that the fourth new animal joins a group of two animals?
Pick:
A) 1/3
B) 1/2
C) 5/9
D) 5/8
E) none of the previous choices
After the first new animal arrives there will be either three groups of 2 animals or two groups   of 3 animals. After the second new animal arrives, in both cases we will end up with two groups   of 2 animals and one group of 3 animals. The third new animal will either A: join a group of two   animals, or B: join a group of three animals. Event A happens with probability 2=3 and we end up   with one group of 2 animals and two groups of 3 animals; Event B happens with probability 1=3   and we end up with four groups of 2 animals. If event A happens, the fourth new animal joins a   group of 2 animals with probability 1=3. If event B happens, the fourth animal joins a group of 2   animals with probability 1. The probability of the fourth new animal joining a group of 2 animals   is (2=3)(1=3) + (1=3)(1) = 5=9.   correct answer 5/9
<result>C</result>
In a certain habitat, animals live in groups of two and groups of three. Each time a new animal
arrives at the habitat, it randomly chooses a group. If the chosen group has two animals in
it, the new animal joins the group forming a group of three animals. If the chosen group has
three animals in it, the new animal takes one of the animals away from the group of three and
forms a separate group of two animals.
If initially there are five animals in the habitat, and new animals arrive one at a time, what is
the probability that the fourth new animal joins a group of two animals?