Question

Which of three members should form the team for the International competition?

• A. 4, 11, 14
• B. 2, 8, 14
• C. 1, 6, 12
• D. 13, 14, 15
• E. 1, 3, 4
• A. 4, 11, 14
• B. 1, 4, 11
• C. 4, 5, 6
• D. 4, 11, 13
• J. Any team can win the competition
• A. 1, 7, 4
• B. 8, 9, 10
• C. 2, 8, 14
• D. 3, 6, 1
• O. Any of remaining students, as it would not matter

Answer 1

The Correct Option is B

Idea. For the International competition the win probability is very low (0.05), so to maximize the small chance of winning we should prioritize students with the highest peak performance (most 100% scores), rather than the highest average (which rewards consistency more than spikes).
Step 1: Rank by number of 100% scores.
From the table (out of 100 tests each): - \#14: 20, \#2: 15, \#8: 12, \#6: 10, \#3/\#10: 8, \#1: 7, \#5/\#12: 6, ...
Step 2: Pick top three.
Top three peak performers are students 14 (20), 2 (15), and 8 (12).
\[ \boxed{\text{International team: Students 2, 8, 14}} \]

Answer 2

Step 1: Probability of winning at district level.
The probability of winning the district competition is very high at 0.95. This means even average-performing students can secure a win. Hence, the selection for district level does not require the very best students.
Step 2: Reserve strong candidates for higher-level competitions.
Since the international and national competitions have much lower probabilities of winning, the strongest students (like 2, 8, 14 for international and 1, 3, 6 for national) should be reserved for those.
Step 3: Choose weaker performers for district level.
Students 4, 11, and 13 all have average = 70 but very low cent-per-cent scores (1, 1, and 2 respectively). Their consistency is decent, but they lack the peak brilliance of others. This makes them ideal for the district team since even average performance can secure the win. \[ \boxed{\text{District team: Students 4, 11, 13}} \]

Answer 3

Step 1: Exclusion of international team.
From Q33, the international team was chosen as \{2, 8, 14\}. These students are no longer eligible for the national competition.
Step 2: Strategy for national competition.
The probability of winning the national contest is only 0.10 (low, but higher than international). To maximize chances, the team should still balance both average performance Consistency) and peak performance (number of cent-per-cent scores).
Step 3: Candidate pool after exclusion.
Remaining high performers Based on averages and cent-per-cent scores): - Student 1: Avg = 70, Cent = 7
- Student 3: Avg = 65, Cent = 8
- Student 6: Avg = 65, Cent = 10
Other students (like 4, 7, 9, 10, 11, 12, 13, 15) have lower peak or less consistency compared to these three.
Step 4: Best team selection.
Students 1, 3, and 6 together provide: - High averages (65–70 rang(E).
- Strong cent-per-cent spikes (7–10 times).
This maximizes the chance of outperforming at the national level.
\[ \boxed{\text{National team: Students 3, 6, 1}} \]