Complete a conditional tree and calculate intersection, total and conditional probabilities.

Complete a conditional tree and calculate intersection, total and conditional probabilities.

10 marks
ProbabilityQuestionBank.pdf

The following probabilities were found for two events \(R\) and \(S\): \(P(R)=\frac13\), \(P(S\mid R)=\frac45\), \(P(S\mid R\prime)=\frac14\).

\(\frac{1}{3}\)
\(R\)
________
\(R'\)
________
\(S\)
________
\(S'\)
\(\frac{1}{4}\)
\(S\)
________
\(S'\)
  1. Copy and complete the tree diagram.[3]
  2. Find the following probabilities.[7]
    1. \(P(R ∩ S)\).
    2. \(P(S)\).
    3. \(P(R | S)\).

Solution:

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Answer:\(P(R\cap S)=\frac4{15}\), \(P(S)=\frac4{15}+\frac16=\frac{13}{30}\), and \(P(R\mid S)=\frac{4/15}{13/30}=\frac8{13}\).

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