Gerolamo Cardano in his book, The Gambling Scholar, written in the early 1500s, considers the following carnival game. There are six dice. Each of the dice has five blank sides. The sixth side has a number between 1 and 6---a different number on each die. The six dice are rolled and the player wins a prize depending on the total of the numbers which turn up. Find, as Cardano did, the expected total without finding its distribution.

• 3
• 3.25
• 3.5
• 3.75
• 4

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Solution: C

The expected total for an individual dice with number [math]i[/math] on its non-blank side is [math]i/6 [/math] and therefore the expected aggregate total equals

[[math]] \sum_{i=1}^6 \frac{i}{6} = 3.5. [[/math]]