I've been cracking my brain about this one for some time now. I've just started this statistics course. The problem is:
A four sided dice is rolled once and then rolled as many times to not get the same number again. The result of the throw is $(k_1,k_2)$, where $k_1$ is the result of the first roll and $k_2$ is the result of the second. All results are equally likely. What is the probability of:
a) $k_1$ is even
b) $k_1$ and $k_2$ are even
c) $(k_1 + k_2) < 5$
How would I go around setting this up? I've really got no techiques up my sleeve.
Edit: my answers so far are
a) 3/8, b) 1/8, c) 1/4 or a) 1/2, b) 1/6, c) 1/3
Am I going about this right?
Best Answer
The dice has 4 sides (1,2,3 and 4), so there are 2 even sides and 2 odd sides. We rolling the dice two times with the limitation, that the first and the second throw shows different results. So there are the following possible results: (1,2), (1,3), (1,4), (2,1), (2,3), (2,4), (3,1), (3,2), (3,4), (4,1), (4,2) and (4,3). Since all possibilities are equally likely, your questions (a,b and c) are simple counting questions.
a) There are 6 out of 12 possibilities, where $k_{1}$ is even $\Rightarrow\;p=\frac{1}{2}$
b) There are 2 out of 12 possibilities, where $k_{1}$ and $k_{2}$ are even $\Rightarrow\;p=\frac{1}{6}$
c) There are 4 out of 12 possibilities, where $k_{1}+k_{2}<5\;\Rightarrow\;p=\frac{1}{3}$