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We let be the weight of a random potato from Maria's kitchen garden. We assume that is normally distributed with an mean of 200 grams and a standard deviation of 40 grams. Maria will pick up potatoes. She picks a random potato from the kitchen garden.

Maria picks up 500 random potatoes from the kitchen garden. How many of these potatoes can she expect to weigh at least 300 grams? \begin{aligned}p\left( x\geq 300\right) =p\left( \dfrac{x-\mu}{\sigma}\geq \dfrac{300-200}{40}\right) \\ =p\left( z\geq \dfrac{100}{40}\right) =p\left( z\geq 2,5\right) \\ =1-P\left( Z <2,5\right) =1-0,9938\\ =0,0062=0,0062\cdot 100=0,62\% \\ 0,62\% of 500=\dfrac{500\cdot 0,62}{100}=3,1\end{aligned} So 3 of 500 potatoes weight at least 300 g.

Now if we assume that the 500 potatoes are binomial distributed and the probability that each potato weigh at least 300 g is as before 0.0062.Let Y be stochastic variable representing the number of potatoes that weights at least $300 \,g$ then $P(Y=3)=0.224$ which is very low. Here there is a mismatch between normal and binomial distribution or maybe I am misunderstanding something here.

F.O
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1 Answers1

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$P(Y=3)\approx 0.224$ is the probability of exactly $3$ big potatoes, but clearly you could have more or fewer just by chance. The beginning of the distribution is roughly

     y     P(Y=y)

 0     0.044
 1     0.139
 2     0.216
 3     0.224
 4     0.174
 5     0.108
 6     0.056
 7     0.025
 8     0.009
 9     0.003
10     0.001

and if you calculate the expected number $E[Y]=\sum y\, P(Y=y)$ then you get about $3.1$

Henry
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  • We have only 3 big potatoes according to the normal distribution so how you assume it was by chance and we could get more? – F.O Apr 18 '22 at 12:36
  • @F.O The normal distribution is giving you the probability of about $0.0062$ that a particular potato is big. Maria then picks $500$ potatoes, and anywhere from $0$ to $500$ of these could be big, though lower numbers are more likely since big potatoes are rare. $3.1$ is the expected number of big potatoes (the mean of the distribution) but not the actual number of big potatoes and my table shows the probabilities of different numbers of big potatoes. – Henry Apr 18 '22 at 13:24