Reaching a 100
Goldman Sachs
You roll a dice, and with every roll you keep adding up the roll's face until the total reaches 100 or more. Which appears more often in the rolls: 1 or 6?
Answer
If you roll a dice any number of times, all die's faces are equally likely to appear, as it is a fair die. So what makes this problem different? The fact that we stop when the sum of dice rolls reaches 100 - this must create the assymmetry which makes either 1 or 6 more likely to appear more often.
To see why, try to think of the different ways in which we reach 100. If the last roll that reaches 100 is a 1, then the sum just before must have been 99 - i.e. there is only one way in which 1 is our last roll. If 2 was our last roll, the sum before must have been 98 or 99, so 2 ways this can happen. 3 ways for 3 to be the last number, 4 ways for 4 to be the last etc. This means there's 6 ways in which 6 is the last number, and only 1 way for 1 to be the last. Similar argument applies for the last number to reach a sum of 99, or 98 etc, so 6 has a clear advantage over 1. Hence, 6 likely appears more often in our rolls.