Complete a dependent-events tree, calculate a total probability and apply Bayes conditional probability.

Complete a dependent-events tree, calculate a total probability and apply Bayes conditional probability.

4 marks
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The events \(B\) and \(C\) are dependent, where \(C\) is the event “a student takes Chemistry” and \(B\) is the event “a student takes Biology”. It is known that \(P(C)=0.4\), \(P(B\mid C)=0.6\), and \(P(B\mid C\prime)=0.5\).

\(0.4\)
\(C\)
________
\(C'\)
________
\(B\)
________
\(B'\)
________
\(B\)
________
\(B'\)
Chemistry
Biology
  1. Complete the following tree diagram.
  2. Calculate the probability that a student takes Biology.
  3. Given that a student takes Biology, what is the probability that the student takes Chemistry?

[4]

Solution:

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Answer:\(P(B)=0.4(0.6)+0.6(0.5)=0.54\). Thus \(P(C\mid B)=\frac{0.4(0.6)}{0.54}=\frac49\).

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