Two dice are thrown simultaneously. The probability that the product of the numbers appearing on the top faces of the dice is a perfect square is
2/9
The problem asks us to determine the probability that when two dice are thrown simultaneously, the product of the numbers appearing on their top faces is a perfect square.
Total Outcomes for Two Dice
When a single die is thrown, there are 6 possible outcomes: 1, 2, 3, 4, 5, or 6. Since we are throwing two dice simultaneously, and the outcome of one die does not affect the other, the total number of possible outcomes is found by multiplying the number of outcomes for each die.
Total number of outcomes = (Outcomes on First Die) $\times$ (Outcomes on Second Die)
Total number of outcomes = $6 \times 6 = 36$
These outcomes can be listed as ordered pairs, for example, (1,1), (1,2), ..., (6,6).
Perfect Square Products
Next, we need to identify the favorable outcomes. These are the pairs of numbers from the two dice whose product is a perfect square. A perfect square is a number that is the result of squaring an integer. For example, $1 = 1^2$, $4 = 2^2$, $9 = 3^2$, and so on.
The smallest possible product when throwing two dice is $1 \times 1 = 1$. The largest possible product is $6 \times 6 = 36$. So, we need to find perfect squares between 1 and 36, inclusive.
The perfect squares in this range are: 1, 4, 9, 16, 25, 36.
Let's list all the pairs of numbers (first die, second die) that result in these perfect square products:
- For a product of 1: The only pair is (1, 1).
- For a product of 4: The pairs are (1, 4), (2, 2), (4, 1).
- For a product of 9: The only pair is (3, 3).
- For a product of 16: The only pair is (4, 4).
- For a product of 25: The only pair is (5, 5).
- For a product of 36: The only pair is (6, 6).
We can also use a table to visualize all possible products and identify the perfect squares:
| First Die ↓ / Second Die → | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 1 (Perfect Square) | 2 | 3 | 4 (Perfect Square) | 5 | 6 |
| 2 | 2 | 4 (Perfect Square) | 6 | 8 | 10 | 12 |
| 3 | 3 | 6 | 9 (Perfect Square) | 12 | 15 | 18 |
| 4 | 4 (Perfect Square) | 8 | 12 | 16 (Perfect Square) | 20 | 24 |
| 5 | 5 | 10 | 15 | 20 | 25 (Perfect Square) | 30 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 (Perfect Square) |
By counting the pairs we listed, or the cells marked "(Perfect Square)" in the table, we find the number of favorable outcomes:
- (1, 1)
- (1, 4)
- (2, 2)
- (3, 3)
- (4, 1)
- (4, 4)
- (5, 5)
- (6, 6)
There are 8 favorable outcomes.
Probability Calculation
The probability of an event is calculated using the formula:
Probability (Event) = $ \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} $
Using the values we found:
Probability (Product is a perfect square) = $ \frac{8}{36} $
To simplify the fraction, we find the greatest common divisor (GCD) of 8 and 36, which is 4. Divide both the numerator and the denominator by 4:
$ \frac{8 \div 4}{36 \div 4} = \frac{2}{9} $
Conclusion
The probability that the product of the numbers appearing on the top faces of the two dice is a perfect square is $ \frac{2}{9} $.
Important Questions from Numerical Estimation
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