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The total amount of marbles - $2500$. There is only one red box out of 5 boxes. I have to calculate the probability that the amount of marbles in the red box will be no more than 523. My solution ( I am using Gaussian integral theorem). $F(z)=\frac{1}{2\pi} \cdot \int_\infty^{z} e^{-\frac{t^2}{2}}.$ The values for the $z$ can be found in Gaussian table, maybe some of you are familiar with it? $$F\left(\frac{523-500}{20}\right) - F\left(\frac{0 - 500}{20}\right) = F\left(1,15 \right) - F\left(-25 \right)=F(1,15) = 0,3749.$$ Is my solution correct?

user
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  • What is $F?$ It looks like some kind of distribution function. – Ross Millikan Nov 04 '19 at 17:04
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    It appears you are using a normal approximation to a binomial. Your notation seems a bit nonstandard to me, so it is hard to follow exactly, but it seems an okay approach to an approximation. Your final result however looks off. The probability of being at a zscore of $\frac{23}{20}$ or less would have been $\approx 0.874928$. You seem to have missed adding $0.5$ to your final result. You might have wanted to use continuity correction however and used $523.5$ rather than $523$ to get a better approximation here of $\approx 0.88000$. – JMoravitz Nov 04 '19 at 17:05
  • @ JMoravitz Yes, I have't added the $0,5$ to the result. But Why should I use $523,5$? Just because of the $0,5$ I add later? – user Nov 04 '19 at 17:11
  • The exact value is closer to $0.879638\dots$, which would be about $0.004$ off from your answer and about $0.00036$ off from the answer with the continuity correction. – JMoravitz Nov 04 '19 at 17:11
  • "Why should I use $523.5$?" Because it gives a more accurate approximation. See https://math.stackexchange.com/questions/416150/what-is-continuity-correction-in-statistics and https://en.wikipedia.org/wiki/Continuity_correction – JMoravitz Nov 04 '19 at 17:12

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