Collaboration is very powerful! Suppose you have a group of F students, each of whom independently has 55% chance of correctly answering each true/false question on a test with 25 questions. Unfortunately, unlike the first puzzle, this time they don’t know whether they know or don’t know each answer. What is the minimum number of students which is sufficient to collaborate on the test, so that there is a strategy which results in them achieving an A (at least 90% answers correct) with probability at least half?

Question

Collaboration is very powerful! Suppose you have a group of F students, each of whom independently has 55% chance of correctly answering each true/false question on a test with 25 questions. Unfortunately, unlike the first puzzle, this time they don’t know whether they know or don’t know each answer. What is the minimum number of students which is sufficient to collaborate on the test, so that there is a strategy which results in them achieving an A (at least 90% answers correct) with probability at least half?

Answer 1

To determine the minimum number of students required to collaborate on the test in order to achieve at least a 90% correct answer rate with a probability of at least half, we can use a strategy called the binomial distribution.

In this scenario, each student has a 55% chance of answering each question correctly, and there are 25 questions in total. We need to find the minimum number of students that, when working together, can collectively answer at least 90% of the questions correctly.

To solve this problem, we can calculate the probability of achieving a certain number of correct answers using the binomial distribution formula. The formula is given by:

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

Where:
- P(X = k) is the probability of achieving exactly k correct answers.
- (n choose k) represents the binomial coefficient, which calculates the number of ways to choose k successes (correct answers) from n trials (questions).
- p is the probability of success (answering a question correctly).
- (1-p) represents the probability of failure (answering a question incorrectly).
- k is the number of correct answers we want to achieve.
- n is the total number of questions.

Since we want to find the minimum number of students required to achieve at least 90% correct answers, we need to find the smallest value of k for which the probability P(X >= k) is at least 0.5. P(X >= k) represents the probability of achieving k or more correct answers.

We can use this approach to iterate through different values of k and calculate the corresponding probabilities until we find the smallest k that satisfies P(X >= k) >= 0.5.

Let's do the calculations step by step:

1. First, we need to calculate the individual student's probability of answering a question correctly (p).
p = 0.55 (given in the problem statement).

2. Now, we need to find the minimum value of k that satisfies P(X >= k) >= 0.5.
We start with k = 23 (90% of 25 questions).

3. Calculate the probability P(X >= k) using the binomial distribution formula:
P(X >= k) = 1 - P(X < k) = 1 - P(X <= k-1)

4. Calculate P(X <= k-1):
P(X <= k-1) = sum(P(X = i)) from i = 0 to k-1
P(X <= k-1) = sum((25 choose i) * p^i * (1-p)^(25-i)) from i = 0 to k-1

5. Calculate P(X >= k):
P(X >= k) = 1 - P(X <= k-1)

6. If P(X >= k) >= 0.5, we have found the minimum value of k. Otherwise, increase k by 1 and repeat steps 4-6 until we find the minimum value of k.

Let's calculate the minimum number of students required to achieve at least a 90% correct answer rate.

100

To determine the minimum number of students required to collaborate on the test in order to achieve at least a 90% correct answer rate with a probability of at least half, we can use a strategy called the binomial distribution.