Question

In a board game, you must roll two 6-sided number cubes. You can only start the game if you roll a 3 on at least one of the number cubes. Suppose you rolled the two number cubes 100 times, would you expect at least one 3 more or less than 34 times? Explain.

Answer 1

Prob(at least one 3) = 1 - Prob(no 3 in both dices)

Prob(no 3 in 1 dice) = 5/6

Prob(no 3 in both dices) = 5/6x5/6 = 25/36

so Prob(at least one 3) = 1 - 25/36 = 11/36


in 100 rolls, the expected rolls w at least one 3 is 100 x 11/36 = 30.56

i.e. less than 34 times

Answer 2

The odds of rolling a 3 on a 6-sided number cube is 1 out of 6. The odds of rolling a 3 on either of two 6-sided cubes are 1/3. This is because 1/6 * 2 = 1/3. Therefore, an average of one-third of rolls will have a 3 in them. 1/3 expressed as a decimal is .3 repeating (.3333333 etc) and that rounds to 33 which is less than 34.

Therefore, the answer is less than 34 and the explanation is above.