STAT 1102 Study Guide - Midterm Guide: 5,6,7,8, Venn Diagram, Sport Utility Vehicle

Make a probability distribution for the given set of events. The sums that appear when two fair four-sided dice (tetrahedrons) with sides 1, 2, 3, and 4 are tossed. Complete the probability distribution below.

Two fair dice are tossed, and the up face on each die is recorded.

a. List the 36 sample points contained in the sample space.

b. Assign probabilities to the sample points.

c. Find the probability of observing each of the following events:

       A: {A 3 appears on each of the two dice.}

       B: {The sum of the numbers is even.}

       C: {The sum of the numbers is equal to 7.}

       D: {A 5 appears on at least one of the dice.}

       E: {The sum of the numbers is 10 or more.}

Rules of dice game:

• All players start with 10 points

• A dice is rolled, and if the number on the dice is 1, 2, 3, 4 or 5, this number is added to each player's score. If the number is 6, the score off all players is reduced to zero

• Before the dice is rolled each player is given the choice of continuing or taking their current score as their final score.

• This process is repeated until a 6 is rolled and the game ends. All players still playing at this point receive zero as their final score.

Suppose that you choose to play the first dice roll.

(i) Compute the expected value and standard deviation of your score after the first dice roll.

(ii) On average, do you do worse or better than someone who choose not to play and take 10 as their final score?

Now suppose that a 6 was not rolled and you chooses to continue playing

(iii) On average, what score would you have before the dice is rolled?

(iv) Compute the expected value for your score after the second dice roll.

(v) On average, do you do worse or better than someone who choose to stop playing after 1 dice roll?

(vi) Is there an optimal number of dice rolls you should continue playing for? Explain your reasoning.

(vii) If the rules are changed such that when the number is 2, 3, 4, 5 or 6, this number is added to your score. While if the number is 1, your score is reduced to zero. Discuss how this affects the optimal number of dice rolls you should continue playing for.