2 out of 3 or 5 out of 8 If given a chance to score football goal freekicks, which should be chosen: $2$ out of $3$ kicks or $5$ out of $8$ kicks.
My thoughts: $2$ out of $3$, since probability of winning would be $\dfrac{2}{3} \approx 0.66$
But in case of $5$ out of $8$, probability would be $\dfrac{5}{8} \approx 0.62$.
I'm not sure if I'm correct.
What if the choice is between $2$ out of $3$ and $4$ out of $6$.

We assume (unrealistically) that on any kick we have a certain fixed probability $p$ of succeeding, and that success/failure on the various attempts are independent. 
The probability of at least $2$ out of $3$ is $A$, where 
$$A=3p^2(1-p)+p^3.$$
The probability of at least $5$ out of $8$ is
$B$, where 
$$B=56p^5(1-p)^3+28p^6(1-p)^2+8p^7(1-p)+ p^8.$$
We want to find out for what values of $p$ we have $A\ge B$. There is a little cancellation, but solving the inequality exactly may not be feasible.
Intuitively, if we are very a poor kicker (but don't always miss!), then $A\gt B$. If we are a good kicker, then $B$ is better. For numerical details, one could ask Wolfram Alpha. 
