Amazon spinner probability

Question

This problem was asked in the python hackathon which I had failed but had taken the screenshot of the problem.

The problem was based on amazon spinner probability;

• My efforts: The word and was my problem I don't know how should I take the word and in the problem.

• Confusion: As, there are 6-sections then what did they mean by 'betterlucknexttime' how could the probability of betterlucknext time be $\frac16$ I have no idea of probability and statistics but any hint on how I can solve this problem will be helpful to me.

It was a coding question but I wanted to understand the mathematics behind the problem...

Answer 1

It is mentioned that the word 'and' is the bottleneck, so I answer the first part for you.

I interpret the question as out of the $n$ times, we see all the $3$ outcomes appearing at least once. Let the event that the corresponding outcome is observed at least once be $A, B, C$. As the question is badly phrased, I ignore the numerical value and let $p_i$ be the probability that the outcome is $i$ in each spin

The complement to this would be one of the desired outcome does not appear.

\begin{align} P(A \cap B \cap C) &= 1-P(A^c \cup B^c \cup C^c)\\ &= 1-P(A^c) -P(B^c) - P(C^c) + P(A^cB^c)+P(A^cC^c)+P(B^cC^c)-P(A^cB^cC^c) \end{align}

We have $P(A^c)=(1-p_A)^n, P(A^cB^c)=(1-p_A-p_B)^n, P(A^cB^cC^c)=(1-p_A-p_B-p_C)^n$. The other values can be computed similarly.