There are 5 girls and 6 boys on the student government committee. A subcommittee of 5 students is being selected to plan activities. What is the probability that the subcommittee will have AT LEAST 3 girls?
The answer was supposed to be 181/462 but I was confused on why that was the answer.
To find the probability that the subcommittee will have AT LEAST 3 girls, we can calculate the probability of two cases:
1) The subcommittee has exactly 3 girls.
2) The subcommittee has exactly 4 girls.
3) The subcommittee has exactly 5 girls.
To calculate the probability of each case, we need to determine the total number of possible subcommittees and the number of favorable subcommittees for each case.
1) The subcommittee has exactly 3 girls:
To determine the number of favorable subcommittees, we choose 3 girls out of 5 to be on the subcommittee (5 choose 3) and then choose 2 boys out of 6 (6 choose 2). The formula for combination (n choose r) is n! / (r! * (n-r)!).
Number of favorable subcommittees with 3 girls = (5 choose 3) * (6 choose 2) = (5! / (3! * 2!)) * (6! / (2! * 4!)) = 10 * 15 = 150.
2) The subcommittee has exactly 4 girls:
To determine the number of favorable subcommittees, we choose 4 girls out of 5 (5 choose 4) and then choose 1 boy out of 6 (6 choose 1).
Number of favorable subcommittees with 4 girls = (5 choose 4) * (6 choose 1) = (5! / (4! * 1!)) * (6! / (1! * 5!)) = 5 * 6 = 30.
3) The subcommittee has exactly 5 girls:
To determine the number of favorable subcommittees, we choose all 5 girls from the 5 available.
Number of favorable subcommittees with 5 girls = (5 choose 5) = 1.
Now, to find the total number of possible subcommittees, we choose any 5 students from the total of 11 students (5 girls + 6 boys).
Total number of subcommittees = (11 choose 5) = 11! / (5! * 6!) = 462.
Finally, we can add up the number of favorable subcommittees for each case and divide by the total number of possible subcommittees to find the probability:
Probability = (Number of favorable subcommittees) / (Total number of subcommittees)
= (150 + 30 + 1) / 462
= 181 / 462.
Hence, the probability that the subcommittee will have AT LEAST 3 girls is 181/462.