Hi any help with this problem would be appreciated: Q: say you have an assignmen

Question

Hi any help with this problem would be appreciated:

Q: say you have an assignment which involves one true/false question. For the assignment students split into groups of 3, each student lets the other know of their answer and the most popular answer is chosen (if we assume the students answer false, false, true then the final answer for the students will be false). The probability that the student knows the answer is 2/3 and there are 90 students.

1) when students are collaborating in groups of three, what's the probability that the majority answer of a group is correct?

2) let the random variable Y be the number of students who get the answer correct, find its mean and variance?

3) using chebyshev inequality, bound P(Y >= 50).

Any help to these problems would be greatly appreciated.

Thank you

Explanation / Answer

if the students answer false, false, true then the final answer for the students will be false.

probability that the student knows the answer is 2/3

there are 90 students therefore 90/3 =30 groups.

there are 3 students in each group therefore posssable way of answers is 33 = 27

if the students answer false, false, true then the final answer for the students will be false, this happen in two times.

P(false answer) = 2/ 27 = 0.0741

P(correct answer) = 1 - 0.0741 = 0.9259

2)

let the random variable Y be the number of students who get the answer correct.

According to binomial distribution.

mean = n* p = (2/3)*90 = 60

variance = n*p*(1-p) = 20

P(y > = 50) = 1 - P( y <=49) = 1 - 0.0106 = 0.9894

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