Probability of Distribution of Apples Question. I have encountered this question which was actually assigned to a Biology class (the deadline has passed). It seemed simple at first but as more time passes by I realise how difficult it is. This is it:
"In an orchard $15\%$ of apples are blemished. Apples are packed in boxes of $10$. What is the probability that a box has less that four blemished apples?"
I'm not even sure what kind of sample space would be appropriate - let alone how large it is. Could I perhaps assume that we have one hundred apples and fifteen are blemished (this is perhaps related to Law of Large Numbers)? Then each element is made up ten boxes with apples distributed in different ways?
Perhaps we have to use Maxwell-Boltzmann or Fermi-Dirac or Bose-Einstein distributions?
Any help would be really appreciated.
 A: Actually you can assume that you have a very large number of apples and therefore sampling from them does not affect the percentage of blemished/not-blemished apples. This gives you a constant percentage of picking a blemished apple in each of the independent samplings and therefore you can use the Binomial Distribution. 
Indeed, the number $X$ of blemished apples in a sample (box) of $n=10$ apples is binomially distributed with parameters $n=10$ and $p=0.15$. Therefore, the probability that a box has less than $4$ blemished apples is equal to $$P(X<4)=P(X\le 3)=\sum_{k=0}^{3}\dbinom{10}{k}0.15^k(0.85)^{10-k}=0.95$$
A: Denote by $X$ the number of blemished apples in a box. Then $P(X < 4) = P(X = 0) + P(X=1)+P(X=2)+P(X=3)$ $=0.85^{10} + {10 \choose 1} 0.85^9 0.15 + {10 \choose 2} 0.85^80.15^2 + {10 \choose 3} 0.85^70.15^3 \approx 0.9500$
A: 
Perhaps we have to use Maxwell-Boltzmann or Fermi-Dirac or Bose-Einstein distributions?

You can use the binomial distribution with $n=10$ and $p=0.15$. The answer is
$$
\begin{align}
p(B<4) &= p(B=0) + p(B=1) + p(B=2) + p(B=3)\\
&= {10\choose0}(1-p)^{10}+{10\choose1}(1-p)^9p+{10\choose2}(1-p)^8p^2+{10\choose3}(1-p)^7p^3\\
\end{align}
$$
where $B$ is the number of blemished apples.
