Two dice are rolled. Determine the probability of the following. (“Doubles” means both dice show the same number.) Rolling an even number or doubles
The correct answer and explanation is:
To solve the probability of rolling an even number or doubles when two dice are rolled, we need to first define the events.
Step 1: Total Possible Outcomes
The total number of possible outcomes when two dice are rolled is given by the product of the number of outcomes on each die. Each die has 6 faces, so the total possible outcomes are: 6×6=366 \times 6 = 36
Step 2: Event 1 – Rolling an Even Number
An even number can be rolled on a die by landing on 2, 4, or 6. So, to get an even sum from rolling two dice, we need to find the outcomes where the sum of the numbers on both dice is even. This happens in the following scenarios:
- Both dice show even numbers (2, 4, or 6).
- Both dice show odd numbers (1, 3, or 5).
The number of outcomes where both dice show even numbers:
- There are 3 even numbers on each die (2, 4, 6).
- The number of outcomes where both dice show even numbers is 3×3=93 \times 3 = 9.
The number of outcomes where both dice show odd numbers:
- There are 3 odd numbers on each die (1, 3, 5).
- The number of outcomes where both dice show odd numbers is 3×3=93 \times 3 = 9.
Thus, the total number of favorable outcomes for rolling an even sum is: 9+9=189 + 9 = 18
Step 3: Event 2 – Rolling Doubles
Doubles occur when both dice show the same number. The possible doubles are: (1,1),(2,2),(3,3),(4,4),(5,5),(6,6)(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)
There are 6 outcomes where doubles are rolled.
Step 4: Combined Event (Even Number or Doubles)
The final step is to determine the probability of rolling an even number or doubles. We must account for the overlap between the two events. In this case, doubles are already included in the event of rolling an even sum (since doubles like (2,2), (4,4), etc., contribute to the even sum). Therefore, we do not need to count the doubles twice.
The total number of favorable outcomes is: 18 (even sum outcomes)+6 (doubles)−6 (doubles already included in even outcomes)=18+6=2418 \text{ (even sum outcomes)} + 6 \text{ (doubles)} – 6 \text{ (doubles already included in even outcomes)} = 18 + 6 = 24
Step 5: Probability Calculation
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes: Probability=2436=23\text{Probability} = \frac{24}{36} = \frac{2}{3}
Final Answer:
The probability of rolling an even number or doubles is 23\frac{2}{3}.