Two dice are thrown. Find
(i) the odds in favour of getting the sum 6
(ii) the odds against getting the sum 7

Explanation

To solve the problem, we will break it down into two parts:

### Part (i): Finding the odds in favor of getting the sum 6.

1. **Total Outcomes When Two Dice Are Thrown**:
- When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes is:
\[ 6 \times 6 = 36 \]

2. **Favorable Outcomes for Getting the Sum 6**:
- We need to find the combinations of the two dice that give a sum of 6. The possible pairs are:
- (1, 5)
- (2, 4)
- (3, 3)
- (4, 2)
- (5, 1)
- This gives us a total of 5 favorable outcomes.

3. **Calculating Odds in Favor**:
- Odds in favor of an event is calculated as the ratio of the number of favorable outcomes to the number of unfavorable outcomes.
- The number of unfavorable outcomes is:
\[ \text{Unfavorable Outcomes} = \text{Total Outcomes} - \text{Favorable Outcomes} = 36 - 5 = 31 \]
- Therefore, the odds in favor of getting the sum 6 is:
\[ \text{Odds in favor} = \frac{\text{Favorable Outcomes}}{\text{Unfavorable Outcomes}} = \frac{5}{31} \]

### Part (ii): Finding the odds against getting the sum 7.

1. **Favorable Outcomes for Getting the Sum 7**:
- Next, we find the combinations of the two dice that give a sum of 7. The possible pairs are:
- (1, 6)
- (2, 5)
- (3, 4)
- (4, 3)
- (5, 2)
- (6, 1)
- This gives us a total of 6 favorable outcomes.

2. **Calculating Odds Against**:
- Odds against an event is calculated as the ratio of the number of unfavorable outcomes to the number of favorable outcomes.
- The number of unfavorable outcomes is:
\[ \text{Unfavorable Outcomes} = \text{Total Outcomes} - \text{Favorable Outcomes} = 36 - 6 = 30 \]
- Therefore, the odds against getting the sum 7 is:
\[ \text{Odds against} = \frac{\text{Unfavorable Outcomes}}{\text{Favorable Outcomes}} = \frac{30}{6} = 5 \]

### Final Answers:
- (i) The odds in favor of getting the sum 6 are \( \frac{5}{31} \).
- (ii) The odds against getting the sum 7 are \( 5 \).