Multinomial probability calculation

Find probabilities when outcomes fall into more than two categories (e.g. good/satisfactory/poor) using binomial or multinomial approaches.

2 questions

CAIE S1 2013 November Q5
5 On trains in the morning rush hour, each person is either a student with probability 0.36 , or an office worker with probability 0.22 , or a shop assistant with probability 0.29 or none of these.
  1. 8 people on a morning rush hour train are chosen at random. Find the probability that between 4 and 6 inclusive are office workers.
  2. 300 people on a morning rush hour train are chosen at random. Find the probability that between 31 and 49 inclusive are neither students nor office workers nor shop assistants.
CAIE S1 2011 November Q6
6 Human blood groups are identified by two parts. The first part is \(\mathrm { A } , \mathrm { B } , \mathrm { AB }\) or O and the second part (the Rhesus part) is + or - . In the UK, \(35 \%\) of the population are group \(\mathrm { A } + , 8 \%\) are \(\mathrm { B } + , 3 \%\) are \(\mathrm { AB } +\), \(37 \%\) are \(\mathrm { O } + , 7 \%\) are \(\mathrm { A } - , 2 \%\) are \(\mathrm { B } - , 1 \%\) are \(\mathrm { AB } -\) and \(7 \%\) are \(\mathrm { O } -\).
  1. A random sample of 9 people in the UK who are Rhesus + is taken. Find the probability that fewer than 3 are group \(\mathrm { O } +\).
  2. A random sample of 150 people in the UK is taken. Find the probability that more than 60 people are group A+.