In single cast with two dice the odds against drawing 7 is

Options

A. 5
B. `(1)/(5)`
C. 6
D. `(1)/(6)`

Answer

To solve the problem of finding the odds against drawing a sum of 7 when rolling two dice, we can follow these steps:

### Step 1: Determine the total outcomes when rolling two dice.
When rolling two dice, each die has 6 faces. Therefore, the total number of outcomes when rolling two dice is:
\[ \text{Total Outcomes} = 6 \times 6 = 36 \]

**Hint:** Remember that each die operates independently, so multiply the number of outcomes for each die.

### Step 2: Identify the favorable outcomes for getting a sum of 7.
The combinations that result in a sum of 7 are:
- (1, 6)
- (2, 5)
- (3, 4)
- (4, 3)
- (5, 2)
- (6, 1)

Counting these combinations, we find there are 6 favorable outcomes.

**Hint:** List all the combinations systematically to ensure you find all possible pairs that add up to 7.

### Step 3: Calculate the probability of rolling a sum of 7.
The probability \( P \) of rolling a sum of 7 is given by the ratio of favorable outcomes to total outcomes:
\[ P(\text{sum of 7}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{6}{36} = \frac{1}{6} \]

**Hint:** Use the formula for probability which is favorable outcomes divided by total outcomes.

### Step 4: Calculate the probability of not rolling a sum of 7.
The probability of not rolling a sum of 7 is the complement of the probability of rolling a sum of 7:
\[ P(\text{not sum of 7}) = 1 - P(\text{sum of 7}) = 1 - \frac{1}{6} = \frac{5}{6} \]

**Hint:** Remember that the total probability must equal 1, so subtract the probability of the event from 1 to find the probability of the event not occurring.

### Step 5: Determine the odds against rolling a sum of 7.
Odds against an event are calculated as the ratio of the probability of the event not occurring to the probability of the event occurring:
\[ \text{Odds against} = \frac{P(\text{not sum of 7})}{P(\text{sum of 7})} = \frac{\frac{5}{6}}{\frac{1}{6}} = 5:1 \]

**Hint:** Odds are expressed as a ratio of two probabilities, so ensure you simplify the fraction correctly.

### Final Answer:
The odds against drawing a sum of 7 when rolling two dice are \( 5:1 \).