Formula to get the sum of values from an adjustable spinner Unfortunately, I am not so good in Maths and I am trying to find out a formula to calculate the sum of a spinner with $2,3$ or $4$ spins.
The spinner
The spinner has $6$ numbers (like a $6$ face dice).
Each number is different.
Each spinner sector has a different size (different probability).


The problem
Giving a spinner with $6$ different numbers and different probabilities for each number, how much is the probability to reach at least a number if I spin it $2$ times? Or $4$ times?

In this case, I should calculate the possibility to have a sum of $40$ or more. I am able to simulate it, but it would be nice if anyone can help me devise a formula.
 A: Lets say you have the probability P of being in a state S defined as,
$P(S_2)=0.08$, $P(S_5)=0.40$, $P(S_{10})=0.24$, $P(S_{15})=0.20$, $P(S_{40})=0.07$, $P(S_{50})=0.01$,
Now if you get $50$ or $40$ in the first go, you don't need to spin the wheel anymore.
If you get any other no. you will have to spin again. Now suppose if you get $10$ and in the next spin $40$ you have already reached the sum. But if you get 15 you will again have to spin it and go through the same process.

Forming this mathematically,
Let $M_{1,40}$ be the event of reaching a sum of $40$ in $1st$ spin, then $M_{1,40} = P(S_{40}) + P(S_{50})$.
Similarly, $M_{2,40} = M_{1,40} + [P(S_{2})+P(S_{5})+P(S_{10})+P(S_{15})].[P(S_{40}) + P(S_{50})] $.
And all other calculations would go on like this.

If you want you can come up with a formula for values $A,B,C,D,E$ having probabilities $a,b,c,d,e$ by hand in a similar fashion and use it directly in simulation.
It is evident here that the minimum no. of spins required to always attain a sum of $N$ is $ceil(\frac{N}{min(A,B,C,D,E)})$.
