What is the probability that the sum of the two dice is an odd number?
Understand the Problem
The question is asking for the probability that the sum of two rolled dice is odd, given that at least one die shows an even number. The high-level approach involves determining the possible outcomes of the dice and applying the rules of probability.
Answer
The probability that the sum of the two dice is an odd number is $\frac{2}{3}$.
Answer for screen readers
The probability that the sum of the two dice is an odd number is $\frac{2}{3}$.
Steps to Solve
- Determine the Total Outcomes with Even Numbers To find the total possible outcomes where at least one die shows an even number, we first consider the outcomes of rolling two dice. Each die has 6 faces, giving us a total of (6 \times 6 = 36) outcomes. However, we need to exclude the outcome where both dice show odd numbers (1, 3, 5). The combinations for both dice showing odd are:
- (1, 1), (1, 3), (1, 5),
- (3, 1), (3, 3), (3, 5),
- (5, 1), (5, 3), (5, 5). This totals (3 \times 3 = 9) outcomes. Thus, the total outcomes with at least one even number is: $$ 36 - 9 = 27 $$
- Identify Outcomes with Odd Sums To achieve an odd sum from two dice, one die must be even, and the other must be odd. Given the dice, even numbers are (2, 4, 6) and odd numbers are (1, 3, 5). Possible combinations are:
- Even die is 2: (2, 1), (2, 3), (2, 5) → 3 outcomes
- Even die is 4: (4, 1), (4, 3), (4, 5) → 3 outcomes
- Even die is 6: (6, 1), (6, 3), (6, 5) → 3 outcomes
- Odd die is 1: (1, 2), (3, 2), (5, 2) → 3 outcomes
- Odd die is 3: (1, 4), (3, 4), (5, 4) → 3 outcomes
- Odd die is 5: (1, 6), (3, 6), (5, 6) → 3 outcomes
Total outcomes generating an odd sum: $$ 3 + 3 + 3 + 3 + 3 + 3 = 18 $$
- Calculate the Probability Now, the probability of getting an odd sum given at least one die is even is: $$ P(\text{odd sum} | \text{at least one even}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{18}{27} = \frac{2}{3} $$
The probability that the sum of the two dice is an odd number is $\frac{2}{3}$.
More Information
In a standard six-sided die, there are three even numbers (2, 4, 6) and three odd numbers (1, 3, 5). The total outcomes from rolling two dice can be systematically categorized to identify total possible outcomes and favorable outcomes leading to a specific sum.
Tips
- Failing to correctly count total combinations that include at least one even number.
- Confusing the nature of sums (odd vs even) when manually calculating or visualizing outcomes.
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