A standard pair of six-sided dice is rolled. What is the probability of rolling a sum less than 11 ?

Question

A standard pair of six-sided dice is rolled. What is the probability of rolling a sum less than 1111? Express your answer as a fraction or a decimal number rounded to four decimal places.

Answer 1

P(2<x<11) = 1- P(11) -P(12)

= 1- 2/36 - 1/36

= 1- 3/36

= 1- 1/12

= 11/12 = ,91666...

=about 92% chance of less than 11

= .9167 rounded to 4 decimal places

36 possible outcomes, 1 is 12, 2 are 11

33 are less than 11

33/36 = 11/12

6&6 = 12

6&5 or 5&6 = 11

3 ways to get 11 or 12

4&6,4&4, 5&5 = 10

3 ways to get 10

3&6, 6&3, 4&5, 5&4 = 9

4 ways to get 9

5 ways to get 8

6 ways to get 7

5 ways to get 6

4 ways for 5

3 ways for 4

2 ways for 3

1 way for 2

36 total outcomes

Probability of less than 11 is

33 ways out of 36 = 33/36 = 11/12

Answer 2

Probability Calculation for Dice Rolls

Problem Statement:

Determine the probability of rolling a sum less than 11 with a standard pair of six-sided dice.

Solution Steps:

1. Total Possible Outcomes: There are 6 sides on each die, so for two dice, there are 6 x 6 = 36 possible outcomes.

2. Favorable Outcomes: We need to count the number of outcomes where the sum of the dice is less than 11.

 - Sums of 2 to 10 are favorable (inclusive).

 - For each sum, count the different combinations that can achieve it.

3. Calculate Probability: Probability = (Number of Favorable Outcomes) / (Total Possible Outcomes).

4. Express the probability as a fraction or a decimal rounded to four decimal places.

Implementation of Solution:

Total possible outcomes = 36

Favorable outcomes for sums from 2 to 10:

 - Sum = 2: 1 combination (1+1)

 - Sum = 3: 2 combinations (1+2, 2+1)

 - Sum = 4: 3 combinations (1+3, 2+2, 3+1)

 - Sum = 5: 4 combinations (1+4, 2+3, 3+2, 4+1)

 - Sum = 6: 5 combinations (1+5, 2+4, 3+3, 4+2, 5+1)

 - Sum = 7: 6 combinations (1+6, 2+5, 3+4, 4+3, 5+2, 6+1)

 - Sum = 8: 5 combinations (2+6, 3+5, 4+4, 5+3, 6+2)

 - Sum = 9: 4 combinations (3+6, 4+5, 5+4, 6+3)

 - Sum = 10: 3 combinations (4+6, 5+5, 6+4)

Total favorable outcomes = 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 = 33

Probability = 33 / 36

As a decimal (rounded to four places) = 0.9167

Answer 3

To find the probability of rolling a sum less than 11 with a pair of standard six-sided dice, we can enumerate all possible outcomes and count the favorable outcomes.

The possible outcomes when rolling two six-sided dice are combinations of numbers from 1 to 6 on each die. There are \(6 \times 6 = 36\) total outcomes.

Now, let's consider the favorable outcomes where the sum is less than 11:

- If the sum is 2, there is only one combination: (1, 1).

- If the sum is 3, there are two combinations: (1, 2) and (2, 1).

- If the sum is 4, there are three combinations: (1, 3), (2, 2), and (3, 1).

- If the sum is 5, there are four combinations: (1, 4), (2, 3), (3, 2), and (4, 1).

- If the sum is 6, there are five combinations: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1).

- If the sum is 7, there are six combinations: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1).

- If the sum is 8, there are five combinations: (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2).

- If the sum is 9, there are four combinations: (3, 6), (4, 5), (5, 4), and (6, 3).

- If the sum is 10, there are three combinations: (4, 6), (5, 5), and (6, 4).

Adding up the favorable outcomes, we get \(1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 = 33\).

Therefore, the probability of rolling a sum less than 11 is the ratio of favorable outcomes to total outcomes:

\[ P(\text{sum less than 11}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{33}{36} \]

Simplifying the fraction, we get:

\[ P(\text{sum less than 11}) = \frac{11}{12} \]

So, the probability of rolling a sum less than 11 is \( \frac{11}{12} \).