Understanding the Dice Probability Problem

The question asks for the probability of an event occurring when two dice are thrown simultaneously. Specifically, we need to find the probability that the sum of the numbers shown on the two dice is a multiple of 4.

Determining the Sample Space

When two standard six-sided dice are thrown, each die can land on any number from 1 to 6. The total number of possible outcomes is the product of the outcomes for each die.

• Number of outcomes for the first die = 6
• Number of outcomes for the second die = 6
• Total possible outcomes = \(6 \times 6 = 36\)

The sample space consists of all possible pairs of outcomes, such as (1,1), (1,2), ..., (6,6).

Identifying Favorable Outcomes (Sum is a Multiple of 4)

We need to find the pairs of outcomes where the sum of the numbers is a multiple of 4. The possible sums range from \(1+1=2\) to \(6+6=12\). The multiples of 4 within this range are 4, 8, and 12.

Outcomes for Sum = 4:

The pairs that sum to 4 are:

• (1, 3)
• (2, 2)
• (3, 1)

There are 3 outcomes where the sum is 4.

Outcomes for Sum = 8:

The pairs that sum to 8 are:

• (2, 6)
• (3, 5)
• (4, 4)
• (5, 3)
• (6, 2)

There are 5 outcomes where the sum is 8.

Outcomes for Sum = 12:

The pair that sums to 12 is:

• (6, 6)

There is 1 outcome where the sum is 12.

Total Favorable Outcomes:

The total number of favorable outcomes is the sum of the outcomes for sums 4, 8, and 12.

Total Favorable Outcomes = 3 + 5 + 1 = 9

Calculating the Probability

The probability of an event is calculated using the formula:

$$ \text{Probability} = \dfrac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} $$

Plugging in the values we found:

$$ \text{Probability (Sum is a multiple of 4)} = \dfrac{9}{36} $$

Simplifying the Probability

The fraction \( \dfrac{9}{36} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 9.

$$ \dfrac{9 \div 9}{36 \div 9} = \dfrac{1}{4} $$

Conclusion

The probability of getting a sum that is a multiple of 4 when two dice are thrown simultaneously is \( \dfrac{1}{4} \).