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?
$\frac{12}{190}$
$\frac{33}{95}$
$\frac{62}{95}$
$\frac{178}{190}$
Answer (2)
Calculate the total combinations of choosing 2 students from 20: C ( 20 , 2 ) = 190 .
Calculate the combinations of choosing 2 girls from 12: C ( 12 , 2 ) = 66 .
Calculate the probability of choosing 2 girls: 190 66 .
Calculate the probability of not choosing 2 girls: 1 − 190 66 = 95 62 .
Explanation
Analyze the problem First, let's analyze the problem. We have a group of 8 boys and 12 girls, making a total of 20 students. We want to find the probability that if we randomly select two students, they are not both girls.
Outline the solution To solve this, we can calculate the total number of ways to choose two students from the 20, and then calculate the number of ways to choose two girls. Then we can find the probability of choosing two girls and subtract that from 1 to find the probability of not choosing two girls.
Calculate total combinations The total number of ways to choose 2 students from 20 is given by the combination formula: C ( n , k ) = k ! ( n − k )! n ! In our case, n = 20 and k = 2 , so: C ( 20 , 2 ) = 2 ! ( 20 − 2 )! 20 ! = 2 ! 18 ! 20 ! = 2 × 1 20 × 19 = 190 So there are 190 total possible combinations of choosing two students.
Calculate combinations of two girls Now, let's calculate the number of ways to choose two girls from the 12 girls. Here, n = 12 and k = 2 , so: C ( 12 , 2 ) = 2 ! ( 12 − 2 )! 12 ! = 2 ! 10 ! 12 ! = 2 × 1 12 × 11 = 66 So there are 66 ways to choose two girls.
Calculate probability of two girls The probability of choosing two girls is the number of ways to choose two girls divided by the total number of ways to choose two students: P ( 2 girls ) = Total number of ways to choose 2 students Number of ways to choose 2 girls = 190 66
Calculate probability of not two girls To find the probability that the two students chosen are not both girls, we subtract the probability of choosing two girls from 1: P ( not 2 girls ) = 1 − P ( 2 girls ) = 1 − 190 66 = 190 190 − 66 = 190 124 We can simplify this fraction by dividing both the numerator and denominator by 2: 190 124 = 95 62
State the final answer Therefore, the probability that the students chosen are not both girls is 95 62 .
Examples
In a classroom, you might want to ensure fair representation in a student committee. If you randomly select students, calculating the probability of not selecting only girls (or only boys) helps ensure a balanced group. This is useful in scenarios where diversity or mixed perspectives are desired.
The probability that the two students chosen from a group of 20 (8 boys and 12 girls) are not both girls is 95 62 . This is found by calculating the total combinations of selecting 2 students and the combinations of selecting 2 girls, then using these to determine the probability of not selecting 2 girls. Thus, the answer is 95 62 .
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