Two students from a group of eight boys and 12 girls are sent to represent the school in a parade.

If the students are chosen at random, what is the probability that the students chosen are not both girls?

StartFraction 12 Over 190 EndFraction
StartFraction 33 Over 95 EndFraction
StartFraction 62 Over 95 EndFraction
StartFraction 178 Over 190 EndFraction

1 answer

To find the probability that the two students chosen are not both girls, we first calculate the total number of ways to choose 2 students from the group of 20 students (8 boys and 12 girls).

The total number of ways to choose 2 students from 20 is given by the combination formula:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

where \(n\) is the total number of students and \(r\) is the number of students to choose.

So, we calculate:

\[ \binom{20}{2} = \frac{20!}{2!(20-2)!} = \frac{20 \times 19}{2 \times 1} = 190 \]

Next, we find the number of ways to choose 2 girls from the 12 girls:

\[ \binom{12}{2} = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66 \]

Now, the probability that both students chosen are girls is given by:

\[ P(\text{both girls}) = \frac{\text{Number of ways to choose 2 girls}}{\text{Total ways to choose 2 students}} = \frac{66}{190} \]

To find the probability that the two students are not both girls, we can use the complement rule:

\[ P(\text{not both girls}) = 1 - P(\text{both girls}) = 1 - \frac{66}{190} \]

Calculating this gives:

\[ P(\text{not both girls}) = 1 - \frac{66}{190} = \frac{190 - 66}{190} = \frac{124}{190} \]

To simplify \(\frac{124}{190}\), we divide both the numerator and denominator by their greatest common divisor (GCD). The GCD of 124 and 190 is 62.

\[ \frac{124 \div 62}{190 \div 62} = \frac{2}{95} \]

Finally, we know from the options that \(P(\text{not both girls}) = \frac{62}{95}\).

Therefore, the probability that the students chosen are not both girls is:

\[ \boxed{\frac{62}{95}} \]