Which game is more advantageous? I got a pretty simple problem but I'm not sure about my solution.
Game $A$: We roll a fair die 4 times. If we get the "6" at least one time, we win.
Game $B$: We roll a fair die 8 times. If we get the "6" at least two times, we win.
Which game is more advantageous for us ?
I calculated the first moments which are $\frac{4}{6}$ for game $A$ and $\frac{8}{6}$ for game $B$.
While the expected Value of $B$ is higher than $A$, the probability for $P(X\geq1)$ in game $A$ equals 0.5177 and for game $B$, $P(X\geq2)=0.3953$. This shows that game $B$ is worse than game $A$.
Which solution is right ?
And another question with regard to this problem:
If you multiply the number of trials (here 4 × 2) and the number of minimum successes (here 1 × 2) by a factor $c$ (here 2) why doesn't the probability equal the same number multiplied by $c$ ?
 A: The number $X$ of 6's in 4 trials has $X \sim \mathsf{Binom}(n=4,p=1/6)$ and
he number $Y$ of 6's in 8 trials has $Y \sim \mathsf{Binom}(n=8,p=1/6).$
In R, $P(X \ge 1) = 1 - P(X = 0) = 0.5177.$
1-dbinom(0, 4, 1/6)
[1] 0.5177469

By contrast, $P(Y \ge 2) = 1 - P(X \le 1) = 0.3953,$ smaller than above.
1 - pbinom(1, 8, 1/6)
[1] 0.3953231

By simulation of a million games of each type:
set.seed(1120)
x = replicate(10^6,  sum(sample(1:6, 4, rep=T)==6))
mean(x >= 1)
[1] 0.517721     # aprx 0.5177

y = replicate(10^6,  sum(sample(1:6, 8, rep=T)==6))
mean(y >= 2)
[1] 0.395072      # aprx 0.3953 +/- 0.001
2*sd(y >= 2)/1000
[1] 0.0009777328  # 95% margin of simulation error

In each figure below, the histogram summarizes simulated values and the (centers of) red circles show exact binomial probabilities.

par(mfrow=c(1,2))
 hist(x, prob=T, br = (-1:4)+.5, col="skyblue2")
  points(0:4, dbinom(0:4, 4, 1/6), col="red")
 hist(y, prob=T, br = (-1:8)+.5, col="skyblue2")
  points(0:8, dbinom(0:8, 8, 1/6), col="red")
par(mfrow=c(1,1))

Bottom line: There are only $625$ chances in $1296$ to
lose the first game and $1\,015\,625$ chances in 
$1\,679\,616$ to lose the second.
