Throw a dice $k$ times.
a) What's the probability of getting a '1' or a '2' at least once?
b) Define the random variable X, which denotes the number of throws until we first get a 1 or a 2.
c) What's the expected number of throws in which we'll get a '1' or '2', if we're throwing the dice 10 times total?
My thoughts:
a) Probably the simplest one that's coming to mind is: $P = 1 - (\frac{2}{3})^k$, so 1 minus the probability of only getting 3,4,5,6 in all k throws. But it feels like some combinations of those numbers should also be thrown in there, and that's where I'm stuck.
b) and c) We'll have to use $E(X)$ for c), but I'm not sure whether the X for problem b) will be the same as the one we need for c). I think the random variable X for c) denotes the number of times we'll get a 1 or 2, in 10 throws, so going from 0 to 10, and once again I'm not sure about the probabilities of each.
Your answer for part a) is correct. To throw a 1 or 2 at least once means to not throw only 3's, 4's, 5's and 6's. The order in which you throw them doesn't matter, so no combinations apply.
For the expected value of X, we have to multiply k by the probability of getting it on the k-th throw. $\frac23$ for every non 1 or 2 and $\frac13$ for the final throw.
$$ \sum_{k=1}^\infty k\bigg(\frac23\bigg)^{k-1}\bigg(\frac13\bigg) = 3$$
This answer makes sense considering there is a $1/3$ chance of throwing a 1 or a 2.