Among 20 girls in a class 11 have brown eyes, if 3 girls are selected at random then what is the probability that :
All have brown eyes
No one of them have brown eyes
At most 2 brown eyes
At least 2 brown eyes
Total number of girls in the class: 20
Number of girls with brown eyes: 11
1) Probability that all three girls have brown eyes:
Number of ways to select 3 girls out of 11 with brown eyes: 11C3
Total number of ways to select 3 girls out of 20: 20C3
Probability = 11C3 / 20C3
2) Probability that none of the three girls have brown eyes:
Number of ways to select 3 girls out of 9 without brown eyes: 9C3
Total number of ways to select 3 girls out of 20: 20C3
Probability = 9C3 / 20C3
3) Probability that at most 2 girls have brown eyes:
This includes the cases where 0, 1, or 2 girls have brown eyes.
Number of ways to select 0 girls with brown eyes: 9C3
Number of ways to select 1 girl with brown eyes: 11C1 * 9C2
Number of ways to select 2 girls with brown eyes: 11C2 * 9C1
Total number of ways to select 3 girls out of 20: 20C3
Probability = (9C3 + 11C1 * 9C2 + 11C2 * 9C1) / 20C3
4) Probability that at least 2 girls have brown eyes:
This includes the cases where 2 or 3 girls have brown eyes.
Number of ways to select 2 girls with brown eyes: 11C2
Number of ways to select 3 girls with brown eyes: 11C3
Total number of ways to select 3 girls out of 20: 20C3
Probability = (11C2 + 11C3) / 20C3