Inside the TA's brain
Recent studies have shown that some small area in our brains is sensitive to nonsensical structures: the 1,000,000 neurons of that area seem to be firing whenever the individual is presented with such structures. This area is now known as the Grader's Area.
In a now famous series of experiments (Pseudoneurologica, 28(1):123-129, 1999), several 6.042 TAs have been presented with students' solutions to exam problems. On reading a solution, the expected number of neurons to fire in the Grader's Area was estimated to be 550,000. Assume that this estimate is indeed correct.
Moreover, a TA realizes that a solution is bad if the number of neurons in his/her Grader's Area that fire is greater than the number of neurons that do not by 200,000.
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What is the least number of neurons in the Grader's Area that must fire in order for the TA to realize that a solution is bad?
In order for the neurons that fire to be at least 200,000 more than those that do not fire, they should be at least 100,000 more than half the total number of neurons. That is: 500,000 + 100,000 = 600,000. -
With the information given, provide the best bound on the probability that the TA realizes a solution is bad. Please answer as a fraction of the form x/y.
Let \(R\) be the number of neurons that fire. Then \(E[R]=550,000\) and a bad solution is recognized iff \(R\geq 600,000\). Therefore, the probability we want is \(\Pr[R\geq 600,000]\). We are not given the variance, so we cannot use Chebyshev. By Markov's Theorem, \[\Pr[R\geq 600,000] \leq \frac{E[R]}{600,000} = \frac{550,000}{600,000} = \frac{11}{12}.\] -
In a follow-up study the standard deviation of the number of neurons that fire on seeing a bad solution was found to be 25,000. Using this additional information, provide a better bound on the probability that the TA realizes a solution is bad. Please answer as a fraction of the form x/y.
Let \(R\) be as above. Then the new study says that \(Var[R]=(25,000)^2\). According to Chebyshev, \[\Pr[|R - E[R]|\geq x] \leq \frac{Var[R]}{x^2} = \]We want to upper bound the probability that \(R \geq 600,000\). Equivalently, this is the probability that \(R - 550,000 \geq 50,000\). The way to use Chebyshev's Theorem is to plug in \(x = 50,000\). Then, we get \[Pr[|R-550,000| \geq 50,000] \leq \frac{25,000^2}{50,000^2} = \frac{1}{4}.\]
Note that this is really an upper bound for the probability that
\(R \geq 600,000\) or \(R \leq 500,000\). There are "one-sided" versions of Chebyshev Bounds, which we omit, that give a slightly tighter bound for the probability of just \(R \geq 600,000\).