Question

Define the sample space. To get simpler expressions, you can prefer the set notations.
(a) (6 Points) What is the sample space of choosing a real number from interval [0, 1] which is larger than 0.5. Specify the properties of the sample space. Design two different random variables for this task.
(b) (6 Points) You play rock-paper-scissors with your friend. The one with total three wins or two subsequent wins will be the winner of the game. Give a probability tree of you being the winner. State the sample space by using probability tree diagram.
(You can use nested probability trees to make the representation easier. For example, you can generate a tree only for win-loss possibilities (without any information about rock-paper-scissors), one another tree for Win, including all possible cases with rock- paper-scissors, similarly one another tree for Loss, again including all possible cases with rock-paper-scissors)

Answer 1

(a) Sample space is the set of all possible outcomes of an experiment. The sample space of choosing a real number from interval [0, 1] which is larger than 0.5 is (0.5, 1].

Let be random variable that take value 1 if and takes value 0 otherwise.

Let be another random variable that take value 2 if and takes value 1 otherwise.

(b) Consider the following tree from the point of view of player A. The nodes are the games played. The left edge says the player A wins and the right edge says that player B wins. We can see there are 10 possibilities of end of the game. Out of which five times player A wins and five times player B wins. That is probablity of winning for any player is 0.5 .

We can write the Sample space in terms of wins and losses for player A.

Sample Space = { WW, WLWW, WLWLW, WLWLL, WLL, LWW, LWLWW, LWLWL, LWLL, LL }

We can also replace W by A and L by B, indicating that the respective player wins.

SS= {AA, ABAA, ABABA, ABABB, ABB, BAA, BABAA, BABAB, BABB, BB }