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Completing a two-stage tree for transport choice and punctuality, with joint, total and reverse conditional probabilities

Completing a two-stage tree for transport choice and punctuality, with joint, total and reverse conditional probabilities

10 marks

On a working day, a commuter travels to work by bus with probability \(0.55\) and otherwise by train. When the commuter takes the bus, the probability of arriving on time is \(0.75\). When the commuter takes the train, the probability of arriving on time is \(0.9\). One working day is chosen at random.

  1. Copy and complete the following tree diagram.[3]
\(0.55\)
Bus
________
Train
\(0.75\)
On time
________
Late
________
On time
________
Late
    1. Find the probability that the commuter takes the bus and arrives on time.
    2. Find the probability that the commuter arrives on time.
    3. Given that the commuter arrives on time, find the probability that the commuter took the bus.[7]

Solution:

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Answer:(a) \(\mathrm{P}(\text{Train})=0.45,\ \mathrm{P}(\text{Late}\mid\text{Bus})=0.25,\ \mathrm{P}(\text{On time}\mid\text{Train})=0.9,\ \mathrm{P}(\text{Late}\mid\text{Train})=0.1\) (b)(i) \(0.4125\) (b)(ii) \(0.8175\) (b)(iii) \(\dfrac{55}{109}\approx0.505\)

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