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18 July, 14:02

You toss three 6-sided dice and record the sum of the three faces facing up. a) Describe precisely a sample space S for this experiment. b) What is the probability that the sum is 16 or more? c) What is the probability the sum is exactly 4 or 5? d) What is the probability the sum is less than 17?

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  1. 18 July, 15:18
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    a.) Sample space means all possible outcomes, since we know the dice are 6 face, then the sample space becomes all possible outcomes when we toss the die.

    b.) 9/216

    c.) 9/216

    d.) 212/216

    Step-by-step explanation:

    Sample space means all possible outcomes, since we know the dice are 6 face, then the sample space becomes all possible outcomes when we toss the die.

    [1,1,1] [1,1,2] [1,1,3] [1,1,4] [1,1,5] [1,1,6]

    [1,2,1] [1,2,2] [1,2,3] [1,2,4] [1,2,5] [1,2,6],

    [1,3,1] [1,3,2] [1,3,3] [1,3,4] [1,3,5] [1,3,6]

    [1,4,1] [1,4,2] [1,4,3] [1,4,4] [1,4,5] [1,4,6]

    [1,5,1] [1,5,2] [1,5,3] [1,5,4] [1,5,5] [1,5,6]

    [1,6,1] [1,6,2] [1,6,3] [1,6,4] [1,6,5] [1,6,6]

    [2,1,1] [2,1,2] [2,1,3] [2,1,4] [2,1,5] [2,1,6]

    [2,2,1] [2,2,2] [2,2,3] [2,2,4] [2,2,5] [2,2,6],

    [2,3,1] [2,3,2] [2,3,3] [2,3,4] [2,3,5] [2,3,6]

    [2,4,1] [2,4,2] [2,4,3] [2,4,4] [2,4,5] [2,4,6]

    [2,5,1] [2,5,2] [2,5,3] [2,5,4] [2,5,5] [2,5,6]

    [2,6,1] [2,6,2] [2,6,3] [2,6,4] [2,6,5] [2,6,6]

    [3,1,1] [3,1,2] [3,1,3] [3,1,4] [3,1,5] [3,1,6]

    [3,2,1] [3,2,2] [3,2,3] [3,2,4] [3,2,5] [3,2,6],

    [3,3,1] [3,3,2] [3,3,3] [3,3,4] [3,3,5] [3,3,6]

    [3,4,1] [3,4,2] [3,4,3] [3,4,4] [3,4,5] [3,4,6]

    [3,5,1] [3,5,2] [3,5,3] [3,5,4] [3,5,5] [3,5,6]

    [3,6,1] [3,6,2] [3,6,3] [3,6,4] [3,6,5] [3,6,6]

    [4,1,1] [4,1,2] [4,1,3] [4,1,4] [4,1,5] [4,1,6]

    [4,2,1] [4,2,2] [4,2,3] [4,2,4] [4,2,5] [4,2,6],

    [4,3,1] [4,3,2] [4,3,3] [4,3,4] [4,3,5] [4,3,6]

    [4,4,1] [4,4,2] [4,4,3] [4,4,4] [4,4,5] [4,4,6]

    [4,5,1] [4,5,2] [4,5,3] [4,5,4] [4,5,5] [4,5,6]

    [4,6,1] [4,6,2] [4,6,3] [4,6,4] [4,6,5] [4,6,6]

    [5,1,1] [5,1,2] [5,1,3] [5,1,4] [5,1,5] [5,1,6]

    [5,2,1] [5,2,2] [5,2,3] [5,2,4] [5,2,5] [5,2,6],

    [5,3,1] [5,3,2] [5,3,3] [5,3,4] [5,3,5] [5,3,6]

    [5,4,1] [5,4,2] [5,4,3] [5,4,4] [5,4,5] [5,4,6]

    [5,5,1] [5,5,2] [5,5,3] [5,5,4] [5,5,5] [5,5,6]

    [5,6,1] [5,6,2] [5,6,3] [5,6,4] [5,6,5] [5,6,6]

    [6,1,1] [6,1,2] [6,1,3] [6,1,4] [6,1,5] [6,1,6]

    [6,2,1] [6,2,2] [6,2,3] [6,2,4] [6,2,5] [6,2,6],

    [6,3,1] [6,3,2] [6,3,3] [6,3,4] [6,3,5] [6,3,6]

    [6,4,1] [6,4,2] [6,4,3] [6,4,4] [6,4,5] [6,4,6]

    [6,5,1] [6,5,2] [6,5,3] [6,5,4] [6,5,5] [6,5,6]

    [6,6,1] [6,6,2] [6,6,3] [6,6,4] [6,6,5] [6,6,6]

    b.) Probability that the sum is 16 or more is

    Pr[4,6,6] + pr[5,5,6] + pr [ 5,6,5] + pr [5,6,6] + pr [6,5,5] + pr [6,5,6] + pr [6,6,4] + pr[6,6,5] + pr [6,6,6]

    Becomes:

    [1/6]³ + [1/6]³ + [1/6]³ + [1/6]³ + [1/6]³ + [1/6]³ + [1/6]³ + [1/6]³ + [1/6]³ = 9/216

    Probability that the sum is 4 or 5 is

    Pr [ 1,1,2] or pr[1,2,2] or pr [1,1,3] or pr [1,2,1] or pr[2,1,2] or pr[1,3,1] or pr[3,1,1] or pr[2,1,1] or pr[2,2,1]

    Becomes:

    [1/6]³ + [1/6]³ + [1/6]³ + [1/6]³ + [1/6]³ + [1/6]³ + [1/6]³ + [1/6]³ + [1/6]³ = 9/216

    Probability that the sum is less than 17

    We take it as:

    1 - probability that the sum is 17 and above.

    Now probability that the sum is 17 and above becomes

    pr[5,6,6] or pr[6,5,6] or pr[6,6,5] or pr[6,6,6]

    = [1/6]³ + [1/6]³ + [1/6]³ + [1/6]³ = 4/216

    Hence, probability that the sum is less than 17 becomes:

    1-4/216 = 212/216.
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