A chocolate factory makes gigantic “death by chocolate” chocolate bars and finds that,
in the long run, approximately 10% of the individual squares contain so much chocolate
to present a health hazard to people consuming them. Every chocolate bar contains 100
squares.
(a) Using the Binomial, Poisson and Normal distributions, write down formulas for
the probability that a single chocolate bar has at least 3 but no more than 7 deadly
squares. ...
Note: Use the expectation of the Binomial distribution for the Poisson distribution.
Use the expectation and the standard deviation of the Binomial distribution for
the Normal distribution. In the formulas based on the Binomial and Poisson
distributions use the Psymbol as shown in the lecture notes. The formula based
on the normal distribution should be an integral.
(b) Use either your calculator or the Scientific Notebook functions BinomialDist, PoissonDist
..(4 marks)

2. Originally Posted by sinyoungoh
A chocolate factory makes gigantic “death by chocolate” chocolate bars and finds that,
in the long run, approximately 10% of the individual squares contain so much chocolate
to present a health hazard to people consuming them. Every chocolate bar contains 100
squares.
(a) Using the Binomial, Poisson and Normal distributions, write down formulas for
the probability that a single chocolate bar has at least 3 but no more than 7 deadly
squares. ...
Note: Use the expectation of the Binomial distribution for the Poisson distribution.
Use the expectation and the standard deviation of the Binomial distribution for
the Normal distribution. In the formulas based on the Binomial and Poisson
distributions use the Psymbol as shown in the lecture notes. The formula based
on the normal distribution should be an integral.
(b) Use either your calculator or the Scientific Notebook functions BinomialDist, PoissonDist
..(4 marks)

Let the probbaility that a square is dangerous be $p$.

Then the number of dangerous squares $N$ has a binomial distribution $B(100,p)$, so the probability that there are at least $3$ but no more than $7$ is:

$P(3\le n \le 7)=b(3;100,p)+b(4;100,p)+b(5;100,p)+b(6;100,p)+b(7 ;100,p)$

Now if $p$ is small we may use the Poisson approximation that the number of dangerous squares has a Poisson distribution with mean $100p$.

And the normal approximation has $N$ is (approximatly) distributed with a normal distribution with mean $100p$, and variance $100p(1-p)$.

Now try it yourself.

RonL

RonL