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Two students from a group of eight boys and twelve 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?

A. [tex]$\frac{12}{190}$[/tex]
B. [tex]$\frac{33}{95}$[/tex]
C. [tex]$\frac{62}{95}$[/tex]
D. [tex]$\frac{178}{190}$[/tex]


Sagot :

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]