We roll two fair 6-sided dice. If a fair dice is rolled, the probability of it landing on any number between [1-6] is equi- probable (i. e., 1 6 ). (a) Find the probability that doubles are rolled. (Rolling doubles mean both dices land in the same number, e. g., 3 and 3.) (b) Given that the roll results in a sum of 6 or less, nd the conditional probability that doubles are rolled. (c) Find the probability that the larger of the two die's outcomes is at least 4.

a.)

b.)

c.)

Step-by-step explanation:

Given - We roll two fair 6-sided dice. If a fair dice is rolled, the probability

             of it landing on any number between [1-6] is equi- probable

             (i.e., ).

To find - (a) Find the probability that doubles are rolled.

               (b) Given that the roll results in a sum of 6 or less, and  

                    the conditional probability that doubles are rolled.

               (c) Find the probability that the larger of the two die's outcomes

                    is at least 4.

Proof -

As given , two fair 6-sided dice is rolled.

The sample space is as follows :

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)  }

Total number of outcomes = 36

a.)

Sample space that double are rolled = { (1, 1), ( 2, 2), ( 3, 3) , ( 4, 4), ( 5, 5), ( 6, 6) }

Total number of possible outcomes = 6

So, Probability = =

b.)

Sample space that roll shown sum of 6 or less = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (4, 1), (4, 2), (5, 1)}

Now,

Total outcomes = 15

Total possible outcome that doubles are rolled = 3 ( i.e (1,1), (2, 2), (3, 3) )

So, Probability =

c.)

Sample space that larger of the two die's outcomes is at least 4 = { (1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (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) }

Total possible outcomes = 27

So, Probability =

y/x=18

step-by-step explanation:

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john wilson is depicted as an extremely intelligent man. the point of the excerpt is to make a sort of distinction between emotional and rational intelligence. he happens to have both attributes, but he clearly favors his intellectual gifts. at the same time, because of his religious background, he's a little bit conflicted with this and it makes him ashamed.

hope this !