Question

Two number cubes are rolled for two separate events: Event A is the event that the sum of numbers on both cubes is less than 10. Event B is the event that the sum of numbers on both cubes is a multiple of 3. List the members of the sample space for event A ∩ B.

Answer 1

We simply have to list the cases that satisfy events A and B.

Note that there are 36 possible outcomes, since both dice can output a number between 1 and 6.

To satisfy event A, we must roll a total of less than 10. The only combinations that don't satisfy this requests are

$(5,5),\ (5,6),\ (6,5)$

So, every other combination satisfies event A.

To satisfy event B, we must roll a total that is a multiple of 3. Here are the possible outcomes, and which rolls lead to them:

A total of 3, obtainable by rolling $(1,2),\ (2,1)$

A total of 6, obtainable by rolling $(1,5),\ (5,1),\ (2,4),\ (4,2),\ (3,3)$

A total of 9, obtainable by rolling $(3,6),\ (6,3),\ (4,5),\ (5,4)$

A total of 12, obtainable by rolling $(6,6)$

Note that all of these combinations satisfy event A as well, except for the last one.

So, events A and B are both satisfied by the following rolls:

$(1,2),\ (2,1),\ (1,5),\ (5,1),\ (2,4),\ (4,2),\ (3,3),\ (3,6),\ (6,3),\ (4,5),\ (5,4),\ (6,6)$