An Answer for Q
The first problem is better known as the Monty Hall Dilemma, named for the host of Let’s Make a Deal. The simple and short answer is this: switch doors
The longer answer, from Education, Mathematics, Fun, Monty Hall Dilemma is this:
There is a 1/3 chance that you’ll hit the prize door, and a 2/3 chance that you’ll miss the prize. If you do not switch, 1/3 is your probability to get the prize. However, if you missed (and this with the probability of 2/3) then the prize is behind one of the remaining two doors. Furthermore, of these two, the host will open the empty one, leaving the prize door closed. Therefore, if you miss and then switch, you are certain to get the prize. Summing up, if you do not switch your chance of winning is 1/3 whereas if you do switch your chance of winning is 2/3.
So it really comes down to an odds thing. Just remember not to recalculate the odds after the first door has been opened. That is, you do not you have a 1/2 chance of winning if you switch doors, you actually have a 2/3.
Now on the second problem, which even I didn’t get. Once again, the short answer is: 1 in 11
To figure this out, simply write out all the possible combinations and add them up:
| Roll # | Die One | Die Two |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 1 | 2 |
| 3 | 1 | 3 |
| 4 | 1 | 4 |
| 5 | 1 | 5 |
| 6 | 1 | 6 |
| 7 | 2 | 1 |
| 8 | 3 | 1 |
| 9 | 4 | 1 |
| 10 | 5 | 1 |
| 11 | 6 | 1 |
There is only one roll (roll #1) that produces a dice combination of 1-1. Since there are 11 possible combinations, you have a 1 in 11 chance.
