Key to "Two fair dice (one red and one green)..."

Alaina Ortiz

Answered question

2022-02-13

Two fair dice (one red and one green) are rolled. What is the probability that the sum is 5, given that the green one is either 4 or 3?

Answer & Explanation

powrotnik5ld

Beginner2022-02-14Added 12 answers

Step 1
Let A be the event that the sum of the two dice is 5
Let B be the event that the green die is either a 3 or 4
Then we want $P (A ∣ B)$ which we calculate using the conditional probability formula:
$P (A ∣ B) = \frac{P (A \cap B)}{P (B)}$
Consider first $P (A \cap B)$ which we can calculate using
$P (A \cap B) = \frac{n (A \cap B)}{n (T)}$
Where, $n (T)$ is the total number of possible outcomes. As there are two dices then each has 6 possible outcomes, making $n (T) = 36$
And, $n (A \cap B)$ is the number of outcomes where the total is 5 and the green die is a 3 or a 4.
If $G = 3 \Rightarrow R - 2$
If $G = 4 \Rightarrow R = 1$
And so $n (A \cap B) = 2$ , and therefore we have:
$P (A \cap B) = \frac{2}{36} = \frac{1}{18}$
Now, let use calculate P(B) in a similar fashion:
$P (B) = \frac{n (B)}{n (T)}$
Where $n (B)$ is the number of outcomes for which the green die is a 3 or a 4. For this outcome we have:
If $G = 3 \Rightarrow R = 1, 2, 3, 4, 5, 6$
If $G = 4 \Rightarrow R = 1, 2, 3, 4, 5, 6$
And so $n (B) = 12$ , and s before $n (T) = 36$ , and so:
$P (B) = \frac{12}{36} = \frac{1}{3}$
And so we can now calculate:
$P (A ∣ B) = \frac{\frac{1}{18}}{\frac{1}{3}} = \frac{3}{18} = \frac{1}{6}$

Katherine Trevino

Beginner2022-02-15Added 9 answers

Step 1
If green is

3

4

and red is

1, 2, 3, 4, 5

6

\begin{array}{cccc} green \\ Sum of & 3 & 4 \\ red & 1 & 4 & 5 \\ 2 & 5 & 6 \\ 3 & 6 & 7 \\ 4 & 7 & 8 \\ 5 & 8 & 9 \\ 6 & 9 & 10 \end{array}

\begin{array}{cccc} green \\ Sum of & 3 & 4 \\ red & 1 & 4 & 5 \\ 2 & 5 & 6 \\ 3 & 6 & 7 \\ 4 & 7 & 8 \\ 5 & 8 & 9 \\ 6 & 9 & 10 \end{array}

As can be seen from the table:
there are a total of 12 possible outcomes and only

2

of those outcomes meet the requirement that the total be

5

So the probability is

\frac{2}{12} = \frac{1}{6}

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