Little math problem(s) February 2019

[pier4r]

And another little one.

A participant has 30 attempts to hit a target.

If they hit a target they get A points (A in {7,8,9,10,12} because it is important compared to the number of attempts), if they hit the target at the next attempt too, they get the amount of points they got at the successful previous attempt plus one.
Therefore in a streak of successful n attempts, they get: A, A+1, A+2, ..., A+(n-1) .
If they an attempt is unsuccessful it ends the streak and the next successful attempt will give again A points.

Problems:
(a): Is the sum of the points collected in the 30 attempts always unique? That is: for a given sum one has only one way to score the points to attain it. In this case I intend independently from failures that do not affect the score.
For example:
Fail, Fail, Success, Success, Fail, Success
Success, Fail, Success, Success, Fail
Fail, Success, Fail, Fail, Fail, Success, Success
are all considered equal as the sum is composed by one streak of two successes and one isolated success.

(b): Is the percentage of successes on 30 attempts is known, is the combination of the sum of the points plus the success percentage unique? That is, for a given sum of points and success percentage one has only one way to attain it. Again here some distributions of failures that do not really affect the score or the success percentage are not considered different.
For example:
Success, Fail, Success, Success, Fail
Fail, Success, Fail, Success, Success
are considered equal as they have one isolated success and one streak of two successes (and have the same success percentage).
while
Fail, Success, Fail, Fail, Fail, Success, Success
is different as the success percentage is different from above.

(c) If the percentage of successes is known, and the final sum of points is known, but the participant didn't necessarily use all 30 attempts, is it possible to determine uniquely how they attained the score?

There is a good possibility I was not clear. In that case please ask for clarification!

Code:


Spoiler alret as usual




















(a) No.
7+8+9+10+11+12+13+14 (one single streak of 8 successes)
or
7+8+9+10+11+12+13+7+7 (one streak with 7 successes and two isolated ones)
give the same amount of points

(b)
(c)