### The psychology of dice

**I.**Suppose that you're facing a die roll in which it's important that you succeed. Maybe your character is on the verge of death, maybe that roll will determine whether your group manages to use diplomacy to sway the villain without needing to fight, or whatever. Now suppose your GM lets you choose one of the following two options:

**A)**Roll a d10 twice. On the first roll, you need to come up with at least a 5, and on the second roll, you need to come up with at least a 6.

**B)**Roll a d10 once. You need to come up with at least an 8.

Which of the two options, if either, do you prefer? Why?

To be clear - the two are mathematically equivalent in that no matter which option you pick, you have the same probability of success. In option a you need to first roll 5 - 10 (60% chance) and then 6-10 (50% chance) for an overall 30% chance, and in the second option you need to roll 8-10 (30% chance). I'm interested in whether people have a psychological preference for one of the options, even if they're explicitly told that the probabilities are the same.

In a similar manner, would you choose to:

**II. A)**Roll a d100 twice. On both occasions, you need to roll 50 or under.

**B)**Roll a d100 once, trying to get 25 or under.

**C)**No preference.

**III. A)**Roll a d100 twice. On both occasions, you need to roll 10 or under.

**B)**Roll a d100 once, trying to roll a 1.

**C)**No preference.

**IV. A)**Roll a d100 thrice, trying to roll 70 or under, 50 or under, and 20 or under.

**B)**Roll a d100 twice, trying to roll 35 or under and 20 or under.

**C)**No preference.

and why?