Probability question. Think its Multinomial.

The Question is:
Packets of a certain brand of shampoo should contain 1kg of shampoo powder. The filling machine for these packets is known to give "fills" (in kg) that are Normally distributed with mean = 1.05 and = 0.04.

a) What proportion of packets have underweight fills (i.e, fills of less than 1.00kg)?

b) if a randomly-selected packet has an underweight fill, what is the probability that the fill weights more than 0.96kg?

c) If I randomly select fout packets from a day's production, what is the probability that

i) exactly one packet has an underweight fill?

ii) at least one packet has an underweight fill?

d) What is the probability that, from these four randomly-chosen packets, I get an average of at least 1.00kg of detergent per packet?

I know you need to use the Multinomial distribution for the first part c) and the z tables for part a) and b) parts but then i get really confused. lol

The joint distribution of Y1, . . . ,Yk is the multinomial
distribution: an important discrete multivariate distribution.
The joint probability function of Y1, . . . ,Yk is
The probability mass function of the multinomial distribution is:

P = [ n! / ( y1! * y2! * ... yk! ) ] * ( p1^y1 * p2^y2 * . . . * pk^yk )

where y = y1 + y2 + . . . + yk.

This generalises the Binomial distribution, with two categories
(‘Success’ and ‘Failure’), to k categories.

ANy help would be much appreciated in advance and will be loved forever.
Last edited:


TS Contributor
But in your question you only have two categories, namely the underweight and
overweight. So your problem just reduced to a binomial distribution problem.