Cumulative probability with changing p

Question

Assuming following situation: I have a 10 sided dice (1-10). I´m allowed to roll the dice 10 times. For the first roll the proability to hit each number is the same 10%. For each roll consecutive roll the chance to roll a 5 is increased by 2%. Now i want to calculate the cumulative probability to hit the 5 exactly two times in 10 tries. I know how to calculate it if the probability to hit the 5 per roll stays constant, but how do i do this if the probability changes with each roll? For the first roll the chance to roll the 5 is 10/100, for the second roll the chance to hit the 5 is 12/100 and so on...

Answer 1

I hope there is a simplier answer than this. I decided to create a speadsheet to solve the problem by brute force.

There are $45$ different outcomes where there is two $5$ on $10$ rolls. Since the probability of getting a $5$ change with each roll of the die, $$P(\text{having $5$ on roll $i$}) = 0.1+0.02*(i-1)$$ I created a spreadsheet to evaluate the probability.

The number are

• the first row is the number of the roll
• the second row is the probability to have a $5$ on that roll
• the first column is just the number of the possibility, it is not use in the computation
• The second and third row are the two rolls where the $5$ appear. E.g. possibility $30$ (row $32$), the $5$ where on roll 4 and roll 10.
• The cells D3 to M47 is the probability associated with the roll. For cell D3, the formula is =SI(OU(D$1=$C3;D$1=$B3);D$2;1-D$2) (my version of Excel is in french, formula might be slightly different for an english version)
• Column N is the product of the probabilities of the 10 rolls
• Finally, the green cell (N2) is the sum of all $45$ possibilities.

The answer is $0.304759152681615\approx 30.5\%$