It is known that 70% of the students staying less than 20 km away from the campus. 80% of them using public transport to travel to the campus. Of those who are staying more than 20 km away from the campus, 10% of them using public transport to travel to the campus. (i) Draw a well-labelled probability tree diagram with the joint probabilities calculated to illustrate this situation. (ii) Find the probability that a randomly selected student using public transport to travel to campus. iii) Find the probability that a randomly selected student staying less than 20 km away from the campus given that he/she does not use public transport to travel to campus. ( iv) Are the events "staying less than 20 km away from the campus" and "using public transport to travel to campus" independent? Explain.
Question
It is known that 70% of the students staying less than 20 km away from the campus. 80% of them using public transport to travel to the campus. Of those who are staying more than 20 km away from the campus, 10% of them using public transport to travel to the campus.
(i) Draw a well-labelled probability tree diagram with the joint probabilities calculated to illustrate this situation.
(ii) Find the probability that a randomly selected student using public transport to travel to campus.
iii) Find the probability that a randomly selected student staying less than 20 km away from the campus given that he/she does not use public transport to travel to campus.
( iv) Are the events "staying less than 20 km away from the campus" and "using public transport to travel to campus" independent? Explain.
Solution
(i) To draw a probability tree diagram, we start with two branches representing the two groups of students: those living less than 20 km from campus (70%) and those living more than 20 km from campus (30%). From each of these branches, we draw two more branches representing the students who use public transport and those who don't. For the group living less than 20 km away, 80% use public transport (0.70.8=0.56) and 20% don't (0.70.2=0.14). For the group living more than 20 km away, 10% use public transport (0.30.1=0.03) and 90% don't (0.30.9=0.27).
(ii) The probability that a randomly selected student uses public transport to travel to campus is the sum of the joint probabilities of students living less than 20 km and more than 20 km away who use public transport. So, P(Public Transport) = 0.56 + 0.03 = 0.59 or 59%.
(iii) The probability that a randomly selected student stays less than 20 km away from the campus given that he/she does not use public transport to travel to campus is the joint probability of students living less than 20 km away who don't use public transport divided by the total probability of students who don't use public transport. So, P(<20km|No Public Transport) = 0.14 / (0.14 + 0.27) = 0.34 or 34%.
(iv) Two events are independent if the probability of one event does not affect the probability of the other event. In this case, the probability of a student using public transport is affected by whether they live less than 20 km away from the campus or not. Therefore, the events "staying less than 20 km away from the campus" and "using public transport to travel to campus" are not independent.
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