This grew too long for a comment, but I think something needs to be cleared up about the question and the answers (+1 to both by the way, both fine answers):
While the answers you have here give probabilities, your question asks about odds, which are not the same thing (See here).
You're no doubt using the terms interchangeably (as is commonly done, and usually understood as many people convert between the two so happily the process isn't even conscious), so both answers would respond to your question. However, I'd like to inject some additional clarity (especially for readers less familiar with moving between the two ways of describing chance events).
Your question actually gives values in terms of one and then the other: "50/50" is effectively odds while "1/3" is probability.
In terms of probability, the events you mention would under random choice have (by symmetry) probabilities $\frac{1}{2}$ and $\frac{1}{3}$. Both answers correctly describe the calculation of the probability of the combined event ("first choice is correct -and- second choice is correct given first choice was correct").
In terms of odds, the two events have odds of 1-1 and 2-1 against (1-2 for), and the combined event has odds of 5-1 against (1-5 for).
So for the sake of completeness (and some clarity for those who might get muddled), you obtain the odds of the combined event by converting any odds of the original events to probabilities, computing the resulting probability (either from first principles by simply counting the events, or using the probability rules for such events, like $P(AB) = P(A)P(B|A)$) -- since we have extensive rules for dealing with probability -- and then you convert the answer back to a statement about odds.