A pair of standard dice are rolled. Find the probability of rolling a sum of 6 with these dice.
[tex]P\left(D_1+D_2=6\right)=\frac{[?]}{}[/tex]
Answer (1)
Determine the total possible outcomes: 6 × 6 = 36 .
Identify favorable outcomes (sum to 6): (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) - a total of 5.
Calculate the probability: 36 5 .
The probability of rolling a sum of 6 is 36 5 .
Explanation
Problem Analysis When rolling two standard dice, we want to find the probability that the sum of the numbers rolled is 6.
Total Possible Outcomes Each die has 6 faces, numbered from 1 to 6. The total number of possible outcomes when rolling two dice is 6 × 6 = 36 .
Favorable Outcomes Now, let's find the combinations of numbers on the two dice that sum to 6. These are:
Die 1 = 1, Die 2 = 5 (1 + 5 = 6)
Die 1 = 2, Die 2 = 4 (2 + 4 = 6)
Die 1 = 3, Die 2 = 3 (3 + 3 = 6)
Die 1 = 4, Die 2 = 2 (4 + 2 = 6)
Die 1 = 5, Die 2 = 1 (5 + 1 = 6)
So, there are 5 combinations that result in a sum of 6.
Calculate the Probability The probability of rolling a sum of 6 is the number of favorable outcomes divided by the total number of possible outcomes: P ( D 1 + D 2 = 6 ) = Total number of outcomes Number of favorable outcomes = 36 5
Therefore, the probability of rolling a sum of 6 with two standard dice is 36 5 .
Final Answer The probability of rolling a sum of 6 with two standard dice is 36 5 .
Examples
Imagine you're playing a board game that requires you to roll two dice and get a sum of 6 to move forward. Knowing the probability of rolling a 6 helps you understand how likely you are to advance on your turn. This calculation is also useful in games of chance, where understanding probabilities can inform your betting strategy.
