Question: Information gathered by a marketing research organization reveals that the cell phones sold by Excellent Communications Corporation under the "Cello" trademark come from four different manufacturers. Manufacturer A supplies 30% of the cell phones while manufacturers B, C and D supply 25%, 25% and 20% respectively. The same information also reveals that 10% of the cell phones supplied by A are substandard, while the corresponding figures for B, C and D are 6%, 10% and 8% respectively.

1. Let X denote the event 'cell phone is substandard'. Also let X' denote the complement of X. Draw a neat tree diagram with all the relevant information clearly written on the branches of the tree. Then compute and explain the meaning of the following probabilities:
   1. \(\Pr \left( C\cap X \right)\) and (ii) \(\Pr \left( D\cap X' \right)\)
      (ii) Now we have that
      \[\Pr \left( D\cap X' \right)=\Pr \left( X'|D \right)\Pr \left( D \right)=\left( 1-0.08 \right)\times 0.20=0.184\]
      This probability represents the probability of choosing a cell phone that was manufactured by D and that is not substandard.
2. Mr. Fox buys a Cello cell phone. What is the probability that it is not substandard?
3. Ms. Radix, a professional engineer, buys a Cello cell phone and finds that it is substandard. What is the probability that this cell phone came out of the production line of manufacturer B?
4. Assume now that the head office of Universal Consulting Engineers buys a set of 15 Cello cell phones. What is the probability that at least two will not be substandard?

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