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Two dice are rolled. List all possible Sample spaces. Taka adding of each possiblities which are get the highest odd?

Mathematics

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1) Two dice are rolled. List all possible sample spaces. When adding up each possibility, which one got the highest total?

  1. List all possible sample spaces when two dice are rolled.

    Each die has 6 faces (1, 2, 3, 4, 5, 6). When two dice are rolled, the sample space SS consists of all ordered pairs (d1,d2)(d_1, d_2), where d1d_1 is the result of the first die and d2d_2 is the result of the second die.

    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)}S = \begin{Bmatrix} (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) \end{Bmatrix}

    There are 6×6=366 \times 6 = 36 possible outcomes in the sample space.

  2. Determine which possibility got the highest total when adding up each possibility.

    The total for each possibility is the sum of the numbers on the two dice (d1+d2d_1 + d_2).
    The minimum sum is 1+1=21+1=2.
    The maximum sum is 6+6=126+6=12.
    The possibility that yields the highest total is when both dice show the highest possible number.
    The highest total is obtained from the outcome (6,6)(6,6), which sums to 6+6=126+6=12.

    The possibility with the highest total is (6,6)\boxed{(6,6)}.

    2) If three dice are tossed, what is the probability of obtaining an even number on all three?

  3. Determine the total number of possible outcomes.

    Each die has 6 possible outcomes (1, 2, 3, 4, 5, 6).
    For three dice, the total number of possible outcomes is 6×6×66 \times 6 \times 6.

    Total outcomes=63=216\text{Total outcomes} = 6^3 = 216
  4. Determine the number of favorable outcomes (even number on all three dice).

    For a single die, the even numbers are {2, 4, 6}. There are 3 even outcomes.
    For three dice to show an even number on all three, each die must land on an even number.

    Favorable outcomes=3×3×3=27\text{Favorable outcomes} = 3 \times 3 \times 3 = 27
  5. Calculate the probability.

    The probability of an event is given by the ratio of favorable outcomes to the total number of possible outcomes.

    P(all three even)=Favorable outcomesTotal outcomes=27216P(\text{all three even}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{27}{216}
  6. Simplify the probability.

    P(all three even)=27216=18P(\text{all three even}) = \frac{27}{216} = \frac{1}{8}

    The probability of obtaining an even number on all three dice is 18\boxed{\frac{1}{8}}.

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