Birthdays On Naboo
200 Nabooan children were born last year and we wish to know how many pairs of them share a birthday.
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Assuming that there are 199 days in a Nabooan year, what is the expected number of pairs of matching birthdays?
Let \(M_{ij}\) be an indicator that students \(i\) and \(j\) have the same birthday. Then \(P=\sum\limits_{1\leq i < j\leq 100} M_{ij}\) is the number of matching pairs. By linearity of expectation, \(E[P]=\sum\limits_{1\leq i < j\leq n}E[M_{ij}]=\binom{200}{2}\cdot\frac{1}{199}=100\). -
What is the variance? Please answer as a fraction of the form x/y.
Since the \(M_{ij}\)'s are pairwise independent, we get that \(Var[P]=\sum\limits_{1\leq i < j\leq 100} Var[M_{ij}]=\binom{100}{2}\cdot\frac{1}{199}\left(1-\frac{1}{199}\right)=\frac{19800}{199}\). -
Using Chebyshev's inequality, give a lower bound of the probability of having between 80 and 120 pairs of matching birthdays. Please answer as a fraction in the form of 1 - x/y.
Using Chebyshev, \(\Pr[|P-E[P]| > 20]\leq\frac{Var[P]}{20^2}=\frac{99}{398}\). Hence, the answer is \(1-\frac{99}{398}\approx 0.25\).