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Completing a conditional tree diagram from three given probabilities, then a joint, a total and a reverse conditional probability

Completing a conditional tree diagram from three given probabilities, then a joint, a total and a reverse conditional probability

10 marks

For two events \(M\) and \(N\), it is known that \(\mathrm{P}(M)=\dfrac37\), \(\mathrm{P}(N\mid M)=\dfrac23\) and \(\mathrm{P}(N\mid M')=\dfrac38\). The tree diagram below represents these events.

________
\(M\)
________
\(M'\)
________
\(N\)
________
\(N'\)
________
\(N\)
________
\(N'\)
  1. Copy the tree diagram and write the probability on each of its six branches.[3]
  2. Find[7]
    1. \(\mathrm{P}(M\cap N)\)
    2. \(\mathrm{P}(N)\)
    3. \(\mathrm{P}(M\mid N)\)

Solution:

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Answer:(a) \(\mathrm{P}(M)=\dfrac37,\ \mathrm{P}(M')=\dfrac47,\ \mathrm{P}(N\mid M)=\dfrac23,\ \mathrm{P}(N'\mid M)=\dfrac13,\ \mathrm{P}(N\mid M')=\dfrac38,\ \mathrm{P}(N'\mid M')=\dfrac58\) (b)(i) \(\dfrac27\) (b)(ii) \(\dfrac12\) (b)(iii) \(\dfrac47\)

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