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If the Blue Lands Down the Die Is Not Rolled Again if the Red Die Find P S T

Tetrahedral Dice

Task

Many games utilise dice which are six-sided and off-white (meaning each face up on the die is as probable to land face). Many games also utilize the sum of ii dice rolled at the aforementioned time to determine movement of game pieces. Notwithstanding, not all dice are half-dozen-sided. Imagine a game in which two off-white four-sided (tetrahedral) die are rolled simultaneously. These dice are in the shape of a pyramid, and when a die is rolled, the consequence is determined by the side that lands face down. Suppose that for these two dice, the possible values (corresponding to the iv sides of the die) that tin be obtained from each die are equally follows:

Die #ane: ane, 2, 3, or 4
Die #2: two, 4, 6, or eight

  1. A certain game determines the motility of players' game pieces based on the SUM of the numbers on the face downward sides when 2 dice are rolled. At that place are ten distinct sum values that can occur, and some of those sums occur more than often than others.
    1. Using an organized list, table, tree diagram, or method of your choosing, develop a list of all 16 possible outcomes (for case, Die #1 = 1 and Die #ii = 2 for a sum of 3; Die #1 = 1 and Die #2 = 4 for a sum of five; and and then on).
    2. From your piece of work in part i, determine the x **distinct sum values** that are possible and calculate the probability of obtaining each sum value. Annotation: equally mentioned to a higher place, some values will occur more than frequently than others.
    3. Using your work in part ii, answer the following questions:
      What is the probability of obtaining a sum of 5?
      What is the probability of obtaining a sum that is more than 5?
      What is the probability of obtaining a sum that is at most five?
      What is the probability of obtaining a sum that is at to the lowest degree v?
      What is the probability of obtaining a sum that is no less than v?
  2. Now consider the instance where the DIFFERENCE in the numbers on the face down sides when ii dice are rolled is important to the game. Unless the two die values are the same (in which case the divergence is 0), the difference for purposes of this game volition always be computed as the larger number value rolled minus the smaller number value rolled. In this style, the divergence value for any whorl of the two dice volition e'er be positive or 0.
    1. Using an organized listing, table, tree diagram, or method of your choosing, develop a listing of all xvi possible outcomes (for case, Die #one = ane and Die #two = two for a difference of 1; Die #1 = ane and Die #2 = iv for a deviation of iii; and so on).
    2. From your work in part i, determine the 8 singled-out difference values that are possible and calculate the probability of obtaining each difference value. Note: as mentioned above, some values will occur more frequently than others.
    3. Using your work in part east, answer the following questions:
      What is the probability of obtaining a departure of v?
      What is the probability of obtaining a difference that is more than than 5?
      What is the probability of obtaining a difference that is less than or equal to five?

IM Commentary

The purpose of this task is to have students develop an organized list, table, etc. to decide all possible outcomes of a chance experiment and then to use this information to calculate various probabilities. Hopefully, students will note that techniques applicative to six-sided dice and sums are too applicative to four-sided dice and differences. With certain approaches, patterns in the listing of possible outcomes will be more easily recognized.

Students will demand to brand distinctions in the language of "more than than" vs. "at least" vs. “at most”, etc. Students should too note that different words can be used to describe the same chemical compound events.

If students are familiar with absolute value, you can also use this terminology in describing the variable in question ii of this task.

Solution

    1. Sum Dice #2
      ii 4 6 8
      Die #one ane 3 5 7 nine
      2 4 6 8 10
      3 5 seven nine eleven
      4 6 eight 10 12
    2. $X$ Probability
      3 $\frac{1}{xvi}$
      4 $\frac{one}{16}$
      5 $\frac{2}{xvi}$
      6 $\frac{2}{16}$
      7 $\frac{two}{sixteen}$
      8 $\frac{2}{16}$
      ix $\frac{2}{16}$
      10 $\frac{ii}{xvi}$
      11 $\frac{one}{sixteen}$
      12 $\frac{1}{xvi}$
    3. What is the probability of obtaining a sum of 5?

    $\frac{2}{16} = \frac18$ (see distribution)

    What is the probability of obtaining a sum that is more than 5?

    This includes all outcomes $X = 6$ to $X = 12$. This is $\frac{12}{16} = \frac34$

    What is the probability of obtaining a sum that is at most 5?

    This includes all outcomes $Ten = iii$ to $Ten = 5$. This is $\frac{4}{16} = \frac14$

    Note: this answer is as well the complement to the previous question.

    What is the probability of obtaining a sum that is at least 5?

    This includes all outcomes $X = five$ to $X = 12$. This is $\frac{14}{16} = \frac78$

    What is the probability of obtaining a sum that is no less than 5?

    This includes all outcomes $X = five$ to $X = 12$. This is $\frac{14}{16} = \frac78$

    Note: this is some other way of saying "at to the lowest degree 5" as in the previous question, hence the similar respond.

    1. Difference Die #2
      ii iv half dozen 8
      Die #1 1 i 3 v vii
      ii 0 ii 4 6
      iii 1 one 3 five
      four two 0 2 4
    2. $Ten$ Probability
      0 $\frac{ii}{16}$
      1 $\frac{3}{16}$
      2 $\frac{three}{16}$
      3 $\frac{ii}{xvi}$
      4 $\frac{ii}{sixteen}$
      5 $\frac{2}{xvi}$
      6 $\frac{1}{sixteen}$
      seven $\frac{1}{16}$
    3. What is the probability of obtaining a sum of 5?

    $\frac{2}{16} = \frac18$ (meet distribution)

    What is the probability of obtaining a sum that is more than 5?

    This includes all outcomes $Ten = half-dozen$ to $X = 7$. This is $\frac{ii}{16}$

    What is the probability of obtaining a sum that is less than or equal to five?

    This includes all outcomes $X = 0$ to $X = 5$. This is $\frac{14}{16}$

    Annotation: this respond is too the complement to the previous question.

Tetrahedral Dice

Many games use dice which are six-sided and fair (meaning each face on the dice is as likely to land face up). Many games also utilise the sum of two dice rolled at the same time to determine move of game pieces. Notwithstanding, not all die are six-sided. Imagine a game in which 2 fair four-sided (tetrahedral) dice are rolled simultaneously. These die are in the shape of a pyramid, and when a die is rolled, the result is adamant by the side that lands face down. Suppose that for these two die, the possible values (corresponding to the four sides of the die) that can exist obtained from each die are as follows:

Die #i: 1, 2, 3, or four
Die #2: two, 4, 6, or viii

  1. A certain game determines the move of players' game pieces based on the SUM of the numbers on the face down sides when 2 die are rolled. There are 10 distinct sum values that can occur, and some of those sums occur more than frequently than others.
    1. Using an organized list, table, tree diagram, or method of your choosing, develop a list of all xvi possible outcomes (for case, Die #i = ane and Die #2 = 2 for a sum of iii; Die #1 = ane and Die #2 = 4 for a sum of 5; so on).
    2. From your work in part i, determine the 10 **distinct sum values** that are possible and calculate the probability of obtaining each sum value. Note: as mentioned in a higher place, some values volition occur more frequently than others.
    3. Using your piece of work in office ii, answer the following questions:
      What is the probability of obtaining a sum of 5?
      What is the probability of obtaining a sum that is more than 5?
      What is the probability of obtaining a sum that is at most 5?
      What is the probability of obtaining a sum that is at to the lowest degree 5?
      What is the probability of obtaining a sum that is no less than 5?
  2. Now consider the case where the Deviation in the numbers on the face down sides when two dice are rolled is important to the game. Unless the two dice values are the same (in which case the difference is 0), the difference for purposes of this game will e'er be computed equally the larger number value rolled minus the smaller number value rolled. In this mode, the departure value for any scroll of the ii dice will always be positive or 0.
    1. Using an organized list, tabular array, tree diagram, or method of your choosing, develop a list of all sixteen possible outcomes (for example, Die #i = 1 and Die #2 = 2 for a difference of 1; Die #ane = 1 and Die #ii = 4 for a difference of 3; and so on).
    2. From your work in office i, determine the 8 distinct departure values that are possible and calculate the probability of obtaining each divergence value. Note: every bit mentioned above, some values will occur more often than others.
    3. Using your work in function due east, respond the post-obit questions:
      What is the probability of obtaining a difference of 5?
      What is the probability of obtaining a difference that is more than v?
      What is the probability of obtaining a difference that is less than or equal to 5?

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Source: https://tasks.illustrativemathematics.org/content-standards/tasks/1410