Complete an alarm/late tree, calculate total probability and apply Bayes conditional probability.

Complete an alarm/late tree, calculate total probability and apply Bayes conditional probability.

11 marks
ProbabilityQuestionBank.pdf

Dumisani is a student at IB World College. The probability that he will be woken by his alarm clock is \(\frac78\). If he is woken by his alarm clock, the probability he will be late for school is \(\frac14\). If he is not woken by his alarm clock, the probability he will be late for school is \(\frac35\). Let \(W\) be the event “Dumisani is woken by his alarm clock” and \(L\) the event “Dumisani is late for school”.

________
\(W\)
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\(W'\)
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\(L\)
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\(L'\)
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\(L\)
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\(L'\)
  1. Copy and complete the tree diagram below.[4]
  2. Calculate the probability that Dumisani will be late for school.[3]
  3. Given that Dumisani is late for school what is the probability that he was woken by his alarm clock?[4]

Solution:

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Answer:\(P(L)=\frac78\frac14+\frac18\frac35=\frac{47}{160}\). Given late, \(P(W\mid L)=\frac{(7/8)(1/4)}{47/160}=\frac{35}{47}\).

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