100 items are checked by 3 controllers. What is probability that each of them will check more than 25 items? Here is full quotation of problem from workbook: "Set of 100 articles randomly allocated to test between the three controllers. Find the probability that each controller has got to test at least 25 articles."
Probability theory. 100 items and 3 controllers
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0anyway, the chosen answer matches with my solution. – 2012-04-15
2 Answers
My try: $\mathbb{P}(N_1\geq 25, N_2\geq 25, N_3\geq 25)=\frac{1}{3^{100}}\sum_{N_1\geq 25, N_2\geq 25, N_3\geq 25|N_1+N_2+N_3=100}\binom {100}{N_1,N_2,N_3}=$ $\frac{100!}{3^{100}}\sum_{N_1}\sum_{N_2}\sum_{N_3}\frac{1}{N_1!N_2!N_3!}=\frac{100!}{3^{100}}\sum_{N_1}\sum_{N_2}\frac{1}{N_1!N_2!(100-N_1-N_2)!}=$ $\frac{100!}{3^{100}}\sum_{N_1=25}^{50}\sum_{N_2=25}^{50-(N_1-25)}\frac{1}{N_1!N_2!(100-N_1-N_2)!}$ You also may be interested in this and this
Let $N_i$ denote the number of items checked by controller $i$. One asks for $1-p$ where $p$ is the probability that some controller got less than $k=25$ items. Since $(N_1,N_2,N_3)$ is exchangeable and since at most two controllers can get less than $k$ items, $p=3u-3v$ where $u=\mathrm P(N_1\lt k)$ and $v=\mathrm P(N_1\lt k,N_2\lt k)$.
Furthermore, $v\leqslant uw$ with $w=\mathrm P(M\lt k)$ where $M$ is binomial $(m,\frac12)$ with $m=75$ hence $w\ll1$. And $N_1$ is binomial $(n,\frac13)$.
Numerically, $u\approx2.805\%$ and $w\approx0.1\%$ hence $1-p\approx1-3u\approx91.6\%$.
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2Yurgen: The answer you were after is really the summation of 925 terms, with factorials and everything? (Which, from my answer, sums roughly to 0.9, but how to see this on the summation?) – 2012-04-15