(Intro to Statistics) A game consists of rolling a pair of dice

[Tasaio]

Hi there,

This question is giving me some trouble...

1.1-6

A game consists of rolling a pair of dice and moving a game piece the number of spaces according to the total number of dots on the dice. In order to move the game piece on a player's first turn, the player must roll a 1 or a 6 on at least one die. Give a sample space for this experiemnt, and list the sample points associated with the event "moving on the first turn."

My attempt

The sample space is:

S = {
(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
}

The event "moving the first turn" is

E = { (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)
}

since the player can only move if a 1 or a 6 is rolled on at least one die.

Should these sets be simplified by the assumption that (x, y) is the same thing as (y, x)?

 

[radou]

...Should these sets be simplified by the assumption that (x, y) is the same thing as (y, x)?

Yes, these are sets, and you should use { , } notation, since they are not ordered pairs.

 

[Tasaio]

Yes, these are sets, and you should use { , } notation, since they are not ordered pairs.

My textbook uses ( , ) for every event within the sample space -- the sample space itself uses { }.

See my original post. My question is, if we have (x, y) and (y, x), where x != y, then should they be considered the same event? We are working with *two* distinct dice.

For example, should (1,6) be considered the same as (6, 1)?

 

[radou]

My textbook uses ( , ) for every event within the sample space -- the sample space itself uses { }.

See my original post. My question is, if we have (x, y) and (y, x), where x != y, then should they be considered the same event? We are working with *two* distinct dice.

For example, should (1,6) be considered the same as (6, 1)?

Yes, since the ordering is not important.

 

[Galileo]

Sorry, but I feel the need to interrupt.

Your sample space contains all possible outcomes of your 'experiment'. In this case you roll two dice. The dice are indeed distinct (you can color them, or maybe you have 1 die and you roll twice). Let's say one is red and the other blue, then the sample space is
$$S=\{(R,B)| R,B \in \{1,2,3,4,5,6\}\}$$
The elements are ordered pairs and order is thus important.

However, since all that is important is whether you have rolled a 1 or 6 you may not be interested in the ordering. In that case you may consider the sample space in which order does not matter. The important thing to realize is that you have to assign a probability to each event (i.e. to each element in your sample space). In the first case each element has a probability of 1/36 of happening. In the second case this is not true. The element {1,2} in the second case can occur in two ways (You can throw a 1 and 2 in two different ways), so the probability of that occurring is 1/18.

You can use either sample space for your problem as long as you take the right probabilities. Both ways will give the same probability of moving on the first turn.