Question

Predict the number of outcomes for the given number of rolls with 2 dice:
Outcome: greater than 6
Number of rolls: 100

Answer 1

To predict the number of outcomes where the sum of two dice is greater than 6 after rolling the dice 100 times, we first need to determine the probability of rolling a sum greater than 6 with a pair of dice.

### Step 1: Calculate the Total Possible Outcomes
Each die has 6 faces, which means when rolling two dice, the total number of outcomes is:
\[
6 \times 6 = 36
\]

### Step 2: Calculate Outcomes Greater Than 6

Next, we need to count how many outcomes give a sum greater than 6. The possible sums from rolling two dice range from 2 to 12. The combinations that result in these sums can be summarized as follows:

- **Sum = 2**: (1, 1) → 1 outcome
- **Sum = 3**: (1, 2), (2, 1) → 2 outcomes
- **Sum = 4**: (1, 3), (2, 2), (3, 1) → 3 outcomes
- **Sum = 5**: (1, 4), (2, 3), (3, 2), (4, 1) → 4 outcomes
- **Sum = 6**: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) → 5 outcomes
- **Sum = 7**: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) → 6 outcomes
- **Sum = 8**: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) → 5 outcomes
- **Sum = 9**: (3, 6), (4, 5), (5, 4), (6, 3) → 4 outcomes
- **Sum = 10**: (4, 6), (5, 5), (6, 4) → 3 outcomes
- **Sum = 11**: (5, 6), (6, 5) → 2 outcomes
- **Sum = 12**: (6, 6) → 1 outcome

#### Count Outcomes Greater Than 6
The sums greater than 6 are 7, 8, 9, 10, 11, and 12. The corresponding outcomes are:
- **Sum = 7**: 6 outcomes
- **Sum = 8**: 5 outcomes
- **Sum = 9**: 4 outcomes
- **Sum = 10**: 3 outcomes
- **Sum = 11**: 2 outcomes
- **Sum = 12**: 1 outcome

Adding these gives:
\[
6 + 5 + 4 + 3 + 2 + 1 = 21 \text{ outcomes}
\]

### Step 3: Calculate Probability of Sum Greater than 6
The probability \( P \) of rolling a sum greater than 6 is:
\[
P(\text{greater than 6}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{21}{36} = \frac{7}{12}
\]

### Step 4: Calculate Expected Outcomes in 100 Rolls
For 100 rolls, the expected number of times the outcome will be greater than 6 is:
\[
E = 100 \times P(\text{greater than 6}) = 100 \times \frac{7}{12} \approx 58.33
\]

Since the expected number must be a whole number, you could say approximately **58 or 59** outcomes would be expected to be greater than 6 in 100 rolls.

### Conclusion
The predicted number of outcomes (sums greater than 6) in 100 rolls of two dice is approximately **58 to 59**.