Question

48. 2 dice are rolled, one black and one white.

How many rolls occur in which the value on the white die is greater than the value on the black die?

Answer #1

2 dice are rolled: one black and one white

Now the total possibilities are 36 as shown below.

(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)

In the values (i,j), let i be the outcome from white die and j be the outcome from black die.

We need to find the cases in which i>j

- There are 6 cases in which the outcomes are same (i=j).
- Remaining 30 cases have equal distribution. In half of them i>j and in other half j>i. So the answer would be 15

We can also see the cases in which i>j

(2,1)

(3,1),(3,2)

(4,1),(4,2),(4,3)

(5,1),(5,2),(5,3),(5,4)

(6,1),(6,2),(6,3),(6,4)(6,5)

So, these are the 15 cases in which i>j

**So, the answer is 15**

1) A 10-sided die is rolled infinitely many times. Let X be the
number of rolls up to and including the first roll that comes up 2.
What is Var(X)?
Answer: 90.0
2) A 14-sided die is rolled infinitely many times. Let X be the
sum of the first 75 rolls. What is Var(X)?
Answer: 1218.75
3) A 17-sided die is rolled infinitely many times. Let X be the
average of the first 61 die rolls. What is Var(X)?
Answer:...

1. A pair of dice is rolled: one is blue and the other one is
white. Find the probability of obtaining:
a. number four on blue AND number five on white.
b. number four on blue OR number five on white.
c.Number two on white AND the sum is odd.
d. Number two on white OR the sum is odd.
e. a sum of 9.
f. a sum of 10.
g. a sum of 11.
h. a sum of 8...

1. A pair of dice is rolled: one is blue and the other one is
white. Find the probability of obtaining:
e. a sum of 9.
f. a sum of 10.
g. a sum of 11.
h. a sum of 8 or more.
i. Both dice have the same number.
j. The dice have different numbers.

Five dice of different colors are being rolled simultaneously. The
numbers that show up on the faces are added up to compute X. How
many different die rolls result in X being divisible by 3?

1. If two dice, one red and one green, are rolled, find the
probability that the red die shows a 3 and the green shows a
six.
2. If two dice are rolled, find the probability that the sum of
the faces of the dice is 7.

Roll two dice, one white and one red. Consider these events:
A: The sum is 7
B: The white die is odd
C: The red die has a larger number showing than the white
D: The dice match (doubles)
Which pair(s) of events are disjoint (events AA and BB are
disjoint if A∩B=∅A∩B=∅)?
Which pair(s) are independent?
Which pair(s) are neither disjoint nor independent?

Suppose two dice (one red, one green) are rolled. Consider the
following events. A: the red die shows 2; B: the numbers add to 2;
C: at least one of the numbers is 5; and D: the numbers do not add
to 8. The numbers do not add to 2.How many elements does it
contain?

Suppose two dice (one red, one green) are rolled. Consider the
following events. A: the red die shows 2; B: the numbers add to 5;
C: at least one of the numbers is 2; and D: the numbers do not add
to 12. Express the given event in symbols. HINT [See Example
5.]
The red die shows 2 and the numbers add to 5.
a) How many elements does it contain?

Two symmetric dice have had two of their sides painted red, two
painted black, one painted yellow, and the other painted white.
When this pair of dice is rolled, what is the
probability both dice land on different
colors?

17. Two dice (one red and one green)
are rolled, and the numbers that face up are observed. Test the
pair of events for independence.
A: The red die is 2, 4, or 5; B: The green die
is even.

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