In a game, a person rolls a wheel with numbers between 1-20.
If he gets a number bigger/including 15 he gets to choose between 3 boxes. 2 of which have a prize and the other one does not. If he chose a winning box, he gets to choose to leave the game or stay. It is known that $70 \%$ stay. If he decides to stay he lets go of the previous prize he won, and gets to choose between 3 other boxes, which have 2 big winning prizes as well and one containing nothing!
Now the questions:
1) Given that a person has not won anything, what is the probability he rolled a number bigger or equal to 18 ?
2) What's the probability a person will earn a big prize?
3) Given a person has won a prize (big or small) what is the probability he chose a winning box?
These are my goes:
1) We know that he has not won anything, and we want to calc $P(n \geq 18 | \text{no win}) = \frac{P(\text{no win}| n \geq 18) \cdot P(n \geq 18)}{P(\text{no win})}$
$P(n \geq 18) = \frac{3}{20} ~~\text{as we have these options: 18, 19, 20}$
$P(\text{no win}) = P(\text{Roll n} \leq 14) + P(\text{Rolling }n\geq15 \text{ and choosing the wrong box} ) + P(\text{Rolling }n\geq15 \text{ and choosing the right box, staying, and choosing the wrong box again} ) = \frac{14}{20} + \frac{6}{20} \cdot \frac{1}{3} + \frac{6}{20} \cdot \frac{2}{3} \cdot 0.7 \cdot \frac{1}{3} = \frac{127}{150}$
$P(\text{no win } | n \geq 18) = \frac{1}{3} + \frac{2}{3} \cdot 0.7 \cdot \frac{1}{3} = \frac{44}{90} = \frac{22}{45}$
And so the final answer for 1 is: $\frac{\frac{22}{45} \cdot \frac{3}{20}}{\frac{127}{150}} = \frac{11}{127}$
2) Big prize is if and only if he chose a number $\geq 15$ , chose the right box, stayed, and chose the right box. So:
$P(\text{big prize}) = \frac{6}{20} \cdot \frac{2}{3} \cdot 0.7 \cdot \frac{2}{3} = \frac{28}{300} = \frac{7}{75}$
3) Well, given he won a prize, it means it is guaranteed he chose a winning box, otherwise he wouldn't be able to keep going in the game, so the probability of this happening is:
$P(\text{choosing winnning box } | \text{won a prize}) = 1$
I would like feedback on my answers, I am not sure about my answers not even about 3. I would appreciate if you could show me the right way as well.
Thank you for reading this whole mess :-)