Question

7.5
score: 10.75/13 answered: 11/13
question 12
a pair of dice, one blue and one green, are rolled and the number showing on the top of each die is recorded. what is the probability that the sum of the numbers on the two dice is 5?
question help: video ebook

Explanation:

Step1: Determine total outcomes

When rolling two dice (blue and green), each die has 6 faces. So total possible outcomes are $6\times6 = 36$.

Step2: Find favorable outcomes

We need pairs $(b, g)$ where $b + g = 5$, $b, g\in\{1,2,3,4,5,6\}$.
Possible pairs: $(1,4)$, $(2,3)$, $(3,2)$, $(4,1)$. So there are 4 favorable outcomes.

Step3: Calculate probability

Probability $P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{4}{36}=\frac{1}{9}$. Wait, no, wait: Wait, let's re - check the favorable outcomes. Wait, when $b = 1,g = 4$; $b = 2,g = 3$; $b = 3,g = 2$; $b = 4,g = 1$. So that's 4? Wait, no, wait, 1 + 4 = 5, 2+3 = 5, 3 + 2=5, 4+1 = 5. So 4 outcomes? Wait, no, wait, 6*6 = 36 total. Wait, but let's list all possible sums:

For two dice, the possible sums and their combinations:

Sum of 2: (1,1) - 1 way

Sum of 3: (1,2),(2,1) - 2 ways

Sum of 4: (1,3),(2,2),(3,1) - 3 ways

Sum of 5: (1,4),(2,3),(3,2),(4,1) - 4 ways

Sum of 6: (1,5),(2,4),(3,3),(4,2),(5,1) - 5 ways

Sum of 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) - 6 ways

Sum of 8: (2,6),(3,5),(4,4),(5,3),(6,2) - 5 ways

Sum of 9: (3,6),(4,5),(5,4),(6,3) - 4 ways

Sum of 10: (4,6),(5,5),(6,4) - 3 ways

Sum of 11: (5,6),(6,5) - 2 ways

Sum of 12: (6,6) - 1 way

So for sum 5, the number of favorable outcomes is 4? Wait, no, wait, 1 + 4, 2+3, 3+2, 4+1: that's 4. So probability is 4/36 = 1/9? Wait, no, wait, 4 divided by 36 is 1/9? Wait, 4/36 simplifies to 1/9? Wait, 4÷4 = 1, 36÷4 = 9. Yes. Wait, but let's check again. Wait, maybe I made a mistake. Wait, 1 and 4: 2 outcomes (blue 1 green 4, blue 4 green 1? No, wait, blue is first, green is second. So (1,4) is blue 1, green 4; (4,1) is blue 4, green 1; (2,3) is blue 2, green 3; (3,2) is blue 3, green 2. So that's 4 outcomes. So probability is 4/36 = 1/9. Wait, but maybe I miscounted. Wait, no, 6*6 = 36 total. So the correct number of favorable outcomes for sum 5 is 4. So probability is 4/36 = 1/9. Wait, but let's check with another approach. The formula for the number of ways to get a sum of $s$ with two dice is $\max(0, s - 1, 7 - s)$? No, better to list. So for $s = 5$, the number of solutions to $x + y=5$, $1\leq x\leq6$, $1\leq y\leq6$ is 4, as we found. So probability is 4/36 = 1/9. Wait, but I think I made a mistake earlier. Wait, no, 4/36 reduces to 1/9. So the probability is 1/9. Wait, but let's check again. Wait, 1+4, 2+3, 3+2, 4+1: that's 4 ordered pairs. So total outcomes 36. So 4/36 = 1/9.

Answer:

$\frac{1}{9}$