Two dice are thrown together and the product of numbers appearing on them is noted. Find the probability that the product is less than 12.

Answer

To solve the problem of finding the probability that the product of the numbers appearing on two thrown dice is less than 12, we can follow these steps:

### Step 1: Determine the Sample Space
When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing two dice is:
\[ \text{Total outcomes} = 6 \times 6 = 36 \]
This means our sample space consists of 36 possible outcomes.

### Step 2: Identify Favorable Outcomes
Next, we need to find the outcomes where the product of the numbers on the two dice is less than 12. We will list all the combinations of the two dice and calculate their products.

- For die 1 showing 1:
- (1,1) → 1
- (1,2) → 2
- (1,3) → 3
- (1,4) → 4
- (1,5) → 5
- (1,6) → 6

- For die 1 showing 2:
- (2,1) → 2
- (2,2) → 4
- (2,3) → 6
- (2,4) → 8
- (2,5) → 10
- (2,6) → 12 (not included)

- For die 1 showing 3:
- (3,1) → 3
- (3,2) → 6
- (3,3) → 9
- (3,4) → 12 (not included)

- For die 1 showing 4:
- (4,1) → 4
- (4,2) → 8
- (4,3) → 12 (not included)

- For die 1 showing 5:
- (5,1) → 5
- (5,2) → 10
- (5,3) → 15 (not included)

- For die 1 showing 6:
- (6,1) → 6
- (6,2) → 12 (not included)

Now, we can summarize the favorable outcomes:
1. (1,1)
2. (1,2)
3. (1,3)
4. (1,4)
5. (1,5)
6. (1,6)
7. (2,1)
8. (2,2)
9. (2,3)
10. (2,4)
11. (2,5)
12. (3,1)
13. (3,2)
14. (3,3)
15. (4,1)
16. (4,2)
17. (5,1)
18. (5,2)
19. (6,1)

Counting these, we find there are 19 favorable outcomes.

### Step 3: Calculate the Probability
The probability \( P \) that the product is less than 12 is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P(\text{product} < 12) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{19}{36} \]

### Final Answer
Thus, the probability that the product of the numbers appearing on the two dice is less than 12 is:
\[ \frac{19}{36} \]

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