A committee of four students will be selected from a list that contains six Grade 9 students and eight Grade 10 students. What is the expected number of Grade 10 students on the committee?
To find the expected number of Grade 10 students on the committee, we need to multiply the probability of selecting each possible number of Grade 10 students by the respective number of Grade 10 students.
Let's calculate the probabilities step by step:
Step 1: Determine the total number of ways to select a committee of four students from a total of 14 (6 Grade 9 and 8 Grade 10) students. This can be calculated using the combination formula:
C(14, 4) = 14!/(4!(14-4)!) = 1001
Step 2: Calculate the probability of selecting 0 Grade 10 students on the committee. Since there are 8 Grade 10 students, the probability of selecting 0 Grade 10 students is:
P(selecting 0 Grade 10 students) = C(6, 4) / C(14, 4) = 15/1001
Step 3: Calculate the probability of selecting 1 Grade 10 student on the committee. This can be calculated as:
P(selecting 1 Grade 10 student) = C(8, 1) * C(6, 3) / C(14, 4) = 48/1001
Step 4: Calculate the probability of selecting 2 Grade 10 students on the committee. This can be calculated as:
P(selecting 2 Grade 10 students) = C(8, 2) * C(6, 2) / C(14, 4) = 168/1001
Step 5: Calculate the probability of selecting 3 Grade 10 students on the committee. This can be calculated as:
P(selecting 3 Grade 10 students) = C(8, 3) * C(6, 1) / C(14, 4) = 168/1001
Step 6: Calculate the probability of selecting 4 Grade 10 students on the committee. This can be calculated as:
P(selecting 4 Grade 10 students) = C(8, 4) * C(6, 0) / C(14, 4) = 70/1001
Now, we can find the expected number of Grade 10 students by multiplying each probability by its corresponding number of Grade 10 students and summing them up:
Expected number of Grade 10 students = (0 * (15/1001)) + (1 * (48/1001)) + (2 * (168/1001)) + (3 * (168/1001)) + (4 * (70/1001))
Finally, calculating the above expression, the expected number of Grade 10 students on the committee is approximately 1.86 (rounded to two decimal places).