Difference between betting X numbers in one wheel-of-luck turn and betting 1 number in X wheel-of-luck turns The scenario is simple: A wheel of luck has 10 numbers and people can bet on any number and win a chocolate if the wheel spins in their number. What is the difference between betting for 5 numbers in _a single_ turn and betting _1_ number in _5_ turns? What is the formula for deciding this?

The probability of success in the case in which you bet $5$ numbers is easy to compute: it's $\frac12$, since there are $10$ numbers and $\frac5{10}=\frac12$.

Now, if you bet _one_ number _five_ times, how can you win? There are thesse possibilities:

* You can win every single time. The odds are $\left(\frac1{10}\right)^5$.
* You can win exactly four times. The odds are $5\times\left(\frac1{10}\right)^4\times\left(1-\frac1{10}\right)$.
* You can win exactly three times. The odds are $10\times\left(\frac1{10}\right)^3\times\left(1-\frac1{10}\right)^2$.
* You can win exactly two times. The odds are $10\times\left(\frac1{10}\right)^2\times\left(1-\frac1{10}\right)^3$.
* You can win exactly once. The odds are $5\times\frac1{10}\times\left(1-\frac1{10}\right)^4$.



So, the probability of success is the sum of these numbers: $0.41761$.