A die is thrown twice . Find the probability that the sum of the two numbers obtains is 5 or 7?

Answer

To solve the problem of finding the probability that the sum of two numbers obtained from throwing a die twice is either 5 or 7, we can follow these steps:

### Step 1: Determine the Sample Space
When a die is thrown twice, each die has 6 faces. Therefore, the total number of outcomes when throwing two dice is:
\[ 6 \times 6 = 36 \]
Thus, the sample space consists of 36 possible outcomes.

### Step 2: Identify Outcomes for Sum of 5
Next, we need to find all the combinations of the two dice that result in a sum of 5. The pairs that yield this sum are:
- (1, 4)
- (2, 3)
- (3, 2)
- (4, 1)

Counting these combinations, we find there are 4 outcomes that result in a sum of 5.

### Step 3: Identify Outcomes for Sum of 7
Now, we will find all the combinations that result in a sum of 7. The pairs that yield this sum are:
- (1, 6)
- (2, 5)
- (3, 4)
- (4, 3)
- (5, 2)
- (6, 1)

Counting these combinations, we find there are 6 outcomes that result in a sum of 7.

### Step 4: Calculate Total Favorable Outcomes
Now, we combine the outcomes for both sums:
- Outcomes for sum of 5: 4
- Outcomes for sum of 7: 6

Total favorable outcomes = 4 + 6 = 10.

### Step 5: Calculate the Probability
The probability of an event is given by the ratio of the number of favorable outcomes to the total number of outcomes in the sample space. Therefore, the probability \( P \) that the sum is either 5 or 7 is:
\[ P(\text{sum is 5 or 7}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{10}{36} = \frac{5}{18} \]

### Final Answer
Thus, the probability that the sum of the two numbers obtained is 5 or 7 is:
\[ \frac{5}{18} \]

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