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# Dice Probabilities Trivia Quiz

### All questions will deal with dice rolls. Be careful - some involve conditional probability! Good Luck!

A multiple-choice quiz by rodney_indy. Estimated time: 4 mins.

Author
rodney_indy
Time
4 mins
Type
Multiple Choice
Quiz #
268,305
Updated
Dec 03 21
# Qns
10
Difficulty
Tough
Avg Score
5 / 10
Plays
1596
Last 3 plays: Guest 136 (0/10), Guest 163 (2/10), Guest 175 (2/10).
This quiz has 2 formats: you can play it as a or as shown below.
Scroll down to the bottom for the answer key.
1. Two six-sided dice are rolled. Find the probability the first die came up a 2 and the second die came up a 5. Hint

1/36
1/12
1/18
1/9

#### NEXT>

2. Two six-sided dice are rolled. Find the probability that one die came up a 2 and the other came up a 6. Hint

1/36
1/18
1/12
1/9

#### NEXT>

3. Two six-sided dice are rolled. Find the probability that at least one die is a 3. Hint

1/3
1/2
11/36
1/6

#### NEXT>

4. Two six-sided dice are rolled. Find the probability that the sum is 8. Hint

1/3
5/36
3/16
5/18

#### NEXT>

5. Two six-sided dice are rolled. Find the probability that the sum is 8 given that the first die was 3. Hint

1/12
1/5
1/6
5/18

#### NEXT>

6. Two six-sided dice are rolled. Find the probability that the first die came up 3 given that the dice sum to 8. Hint

5/18
1/6
1/5
1/12

#### NEXT>

7. Two six-sided dice are rolled. Find the probability that their sum is not 7. Hint

3/4
5/6
2/3
23/36

#### NEXT>

8. Two six-sided dice are rolled. Find the probability that their sum is not 8, given that their sum is not 7. Hint

5/6
2/3
3/4
23/36

#### NEXT>

9. Two six-sided dice are rolled. Find the probability that the first die is not a 5, given that the second die is not a 2. Hint

2/3
5/6
25/36
29/36

#### NEXT>

10. Two six-sided dice are rolled. But this time, the dice aren't fair: For each die, a 1 is twice as likely to be rolled as a 2, a 2 is twice as likely to be rolled as a 3, ..., and a 5 is twice as likely to be rolled as a 6 (in other words, each number is twice as likely as the number that follows it). So what is the probability of rolling a sum of 7? Hint

64/1223
64/1023
64/1123
64/1323

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Quiz Answer Key and Fun Facts
1. Two six-sided dice are rolled. Find the probability the first die came up a 2 and the second die came up a 5.

Answer: 1/36

There are 6 possibilities for the first die and 6 possibilities for the second die. Hence there are 6 * 6 = 36 possible dice rolls. There is only one of them: (2,5) which has 2 on the first die and 5 on the second die. Thus the probability is 1/36.
2. Two six-sided dice are rolled. Find the probability that one die came up a 2 and the other came up a 6.

Answer: 1/18

This event consists of the outcomes (2,6), (6,2). Hence it has probability 2/36 = 1/18.
3. Two six-sided dice are rolled. Find the probability that at least one die is a 3.

Answer: 11/36

The probability that the first die is 3 is 1/6, the probability that the second die is 3 is 1/6, and the probability that both dice are 3 is 1/36. By the principle of inclusion/exclusion, the probability that the first die or the second die is 3 is just 1/6 + 1/6 - 1/36 = 11/36. This probability is useful in backgammon - this is the probability of entering off the bar when there is only one open point in the opponent's table.
4. Two six-sided dice are rolled. Find the probability that the sum is 8.

Answer: 5/36

The outcomes in the event "the sum is 8" are: (2,6), (3,5), (4,4), (5,3), and (6,2). Since there are 5 ways the sum can be 8, the probability is 5/36.
5. Two six-sided dice are rolled. Find the probability that the sum is 8 given that the first die was 3.

Answer: 1/6

This is a conditional probability question. We are given that the first die was 3: (3,1), (3,2), (3,3), (3,4), (3,5), (3,6)

Of these 6 possibilities, (3,5) is the only outcome where the dice sum to 8. So the probability is 1/6.
6. Two six-sided dice are rolled. Find the probability that the first die came up 3 given that the dice sum to 8.

Answer: 1/5

This is another conditional probability question. We are given that the dice sum to 8: (2,6), (3,5), (4,4), (5,3), (6,2)
Of the 5 ways that the dice sum to 8, only one has the first die 3: (3,5). Hence the probability is 1/5.
7. Two six-sided dice are rolled. Find the probability that their sum is not 7.

Answer: 5/6

The ways in which a 7 can be rolled are: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). There are 6 ways a sum of 7 can be rolled. There are 36 total dice rolls, thus there are 30 ways a sum of 7 cannot be rolled. Hence the probability is 30/36 = 5/6.
8. Two six-sided dice are rolled. Find the probability that their sum is not 8, given that their sum is not 7.

Answer: 5/6

We are given that the sum is not 7. In question number 7, we saw that there were 30 ways the sum could not be 7. The ways we can roll a sum of 8 are: (2,6), (3,5), (4,4), (5,3), and (6,2). So among the 30 ways we cannot roll a sum of 7, there are 25 ways we cannot roll a sum of 8. Hence the probability is 25/30 = 5/6.
9. Two six-sided dice are rolled. Find the probability that the first die is not a 5, given that the second die is not a 2.

Answer: 5/6

Let E be the event "the first die is a 5" and F be the event "the second die is a 2." E and F are independent events. Hence so are the complements of E and F. This means that the probability of the complement of E given the complement of F is just the probability of the complement of E, which is 1 - 1/6 = 5/6.
10. Two six-sided dice are rolled. But this time, the dice aren't fair: For each die, a 1 is twice as likely to be rolled as a 2, a 2 is twice as likely to be rolled as a 3, ..., and a 5 is twice as likely to be rolled as a 6 (in other words, each number is twice as likely as the number that follows it). So what is the probability of rolling a sum of 7?

Answer: 64/1323

Let x = the probability of rolling a 6. Then 2x is the probability of rolling a 5, 2*2x = 4x = the probability of rolling a 4, 8x = the probability of rolling a 3, 16x = the probability of rolling a 2, and 32x = the probability of rolling a 1. These are all the outcomes, so these probabilities must sum to 1: x + 2x + 4x + 8x + 16x + 32x = 1 which simplifies to 63x = 1 which means x = 1/63. So Pr(1) = 32/63, Pr(2) = 16/63, Pr(3) = 8/63, Pr(4) = 4/63, Pr(5) = 2/63, and Pr(6) = 1/63. There are six ways to roll a sum of 7:

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

By independence, their corresponding probabilities are

(32/63)*(1/63), (16/63)*(2/63), (8/63)*(4/63), (4/63)*(8/63), (2/63)*(16/63), and (1/63)*(32/63).

Each probability is 32/63^2. We add up these six probabilities to get the probability of rolling a sum of 7: 6*(32/63^2) = 64/1323.

I hope you enjoyed my quiz! Thanks for playing!
Source: Author rodney_indy

This quiz was reviewed by FunTrivia editor crisw before going online.
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