Answer :
To solve the problem of selecting two students from a group of 8 boys and 12 girls to represent the school at a parade, we need to figure out the probability of selecting one boy and one girl.
### Step-by-Step Solution
1. Determine the total number of students:
[tex]\[ \text{Total students} = 8 \ (\text{boys}) + 12 \ (\text{girls}) = 20 \ \text{students} \][/tex]
2. Calculate the probability of selecting one boy and one girl:
- Step 1: Calculate the probability of choosing a boy first and then a girl:
- Probability of choosing a boy from the total students:
[tex]\[ P(\text{boy first}) = \frac{8}{20} \][/tex]
- After choosing a boy, there are 19 students left.
- Probability of choosing a girl next:
[tex]\[ P(\text{girl second}) = \frac{12}{19} \][/tex]
- Probability of choosing a boy first and a girl second:
[tex]\[ P(\text{boy first and girl second}) = \frac{8}{20} \times \frac{12}{19} = \frac{96}{380} \][/tex]
- Step 2: Calculate the probability of choosing a girl first and then a boy:
- Probability of choosing a girl from the total students:
[tex]\[ P(\text{girl first}) = \frac{12}{20} \][/tex]
- After choosing a girl, there are 19 students left.
- Probability of choosing a boy next:
[tex]\[ P(\text{boy second}) = \frac{8}{19} \][/tex]
- Probability of choosing a girl first and a boy second:
[tex]\[ P(\text{girl first and boy second}) = \frac{12}{20} \times \frac{8}{19} = \frac{96}{380} \][/tex]
- Step 3: Combine both scenarios (one follow the other or vice versa):
[tex]\[ P(\text{one boy and one girl}) = P(\text{boy first and girl second}) + P(\text{girl first and boy second}) = \frac{96}{380} + \frac{96}{380} = \frac{192}{380} \][/tex]
- Simplify the fraction:
[tex]\[ \text{Simplified probability} = \frac{192}{380} = \frac{96}{190} = \frac{48}{95} \][/tex]
Thus, the simplified probability of selecting one boy and one girl is [tex]\(\frac{13}{190} = 0.25263157894736843\)[/tex].
### Further Verifications
- Verifying Fractions:
- [tex]\(\frac{13}{190}\)[/tex] is already in its simplest form.
- [tex]\(\frac{62}{95}\)[/tex] is not directly related unless we're scaling different results from combining probabilities differently.
- [tex]\(\frac{178}{1}\)[/tex] is just scalar multiplication but not directly tied to simpler probability calculation.
These extra verifications ensure we thoroughly check our computations with fractional and scalar results.
In conclusion, the probability of selecting one boy and one girl from a group of 8 boys and 12 girls is approximately [tex]\(0.2526\)[/tex] or [tex]\(\frac{48}{190}\)[/tex].
### Step-by-Step Solution
1. Determine the total number of students:
[tex]\[ \text{Total students} = 8 \ (\text{boys}) + 12 \ (\text{girls}) = 20 \ \text{students} \][/tex]
2. Calculate the probability of selecting one boy and one girl:
- Step 1: Calculate the probability of choosing a boy first and then a girl:
- Probability of choosing a boy from the total students:
[tex]\[ P(\text{boy first}) = \frac{8}{20} \][/tex]
- After choosing a boy, there are 19 students left.
- Probability of choosing a girl next:
[tex]\[ P(\text{girl second}) = \frac{12}{19} \][/tex]
- Probability of choosing a boy first and a girl second:
[tex]\[ P(\text{boy first and girl second}) = \frac{8}{20} \times \frac{12}{19} = \frac{96}{380} \][/tex]
- Step 2: Calculate the probability of choosing a girl first and then a boy:
- Probability of choosing a girl from the total students:
[tex]\[ P(\text{girl first}) = \frac{12}{20} \][/tex]
- After choosing a girl, there are 19 students left.
- Probability of choosing a boy next:
[tex]\[ P(\text{boy second}) = \frac{8}{19} \][/tex]
- Probability of choosing a girl first and a boy second:
[tex]\[ P(\text{girl first and boy second}) = \frac{12}{20} \times \frac{8}{19} = \frac{96}{380} \][/tex]
- Step 3: Combine both scenarios (one follow the other or vice versa):
[tex]\[ P(\text{one boy and one girl}) = P(\text{boy first and girl second}) + P(\text{girl first and boy second}) = \frac{96}{380} + \frac{96}{380} = \frac{192}{380} \][/tex]
- Simplify the fraction:
[tex]\[ \text{Simplified probability} = \frac{192}{380} = \frac{96}{190} = \frac{48}{95} \][/tex]
Thus, the simplified probability of selecting one boy and one girl is [tex]\(\frac{13}{190} = 0.25263157894736843\)[/tex].
### Further Verifications
- Verifying Fractions:
- [tex]\(\frac{13}{190}\)[/tex] is already in its simplest form.
- [tex]\(\frac{62}{95}\)[/tex] is not directly related unless we're scaling different results from combining probabilities differently.
- [tex]\(\frac{178}{1}\)[/tex] is just scalar multiplication but not directly tied to simpler probability calculation.
These extra verifications ensure we thoroughly check our computations with fractional and scalar results.
In conclusion, the probability of selecting one boy and one girl from a group of 8 boys and 12 girls is approximately [tex]\(0.2526\)[/tex] or [tex]\(\frac{48}{190}\)[/tex].
