Certainly! Let's solve the problem step by step to find the probability that the two students chosen are not both girls.
### Step 1: Calculate Total Number of Students
There are a total of 8 boys and 12 girls.
[tex]\[
\text{Total students} = 8 + 12 = 20
\][/tex]
### Step 2: Calculate Total Number of Ways to Choose 2 Students
We need to find the number of ways to choose 2 students from a total of 20 students. This can be done using the combination formula:
[tex]\[
\binom{20}{2} = \frac{20!}{2!(20-2)!} = \frac{20 \times 19}{2 \times 1} = 190
\][/tex]
So, the total number of combinations is 190.
### Step 3: Calculate Number of Ways to Choose 2 Girls
Next, we need to find the number of ways to choose 2 girls from the 12 girls.
[tex]\[
\binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66
\][/tex]
So, the number of combinations to choose 2 girls is 66.
### Step 4: Calculate the Probability Both Students are Girls
To find the probability that both students chosen are girls, we divide the number of ways to choose 2 girls by the total number of ways to choose 2 students.
[tex]\[
\text{Probability both are girls} = \frac{\binom{12}{2}}{\binom{20}{2}} = \frac{66}{190} \approx 0.3474
\][/tex]
### Step 5: Calculate the Probability that the Students Chosen are Not Both Girls
The probability that the students chosen are not both girls is 1 minus the probability that both are girls.
[tex]\[
\text{Probability not both are girls} = 1 - \text{Probability both are girls} = 1 - 0.3474 \approx 0.6526
\][/tex]
### Final Answer
To convert the probability to a fraction, we see that:
[tex]\[
0.6526 \approx \frac{124}{190}
\][/tex]
or simplified,
[tex]\[
\frac{124}{190} = \frac{62}{95}
\][/tex]
Therefore, the probability that the students chosen are not both girls is:
[tex]\[
\boxed{\frac{62}{95}}
\][/tex]
