At a certain gas station, 60% of customers use unleaded gas, 35% of customers us

Question

At a certain gas station, 60% of customers use unleaded gas, 35% of customers use premium gas, and the rest use diesel. Of those customers using unleaded gas 70% fill their tank to the full. Of those customers using premium, 30% fill their tank to the full, whereas those using diesel 90% fill their car's tank to the full. Users of gas types and filling of tanks is shown in the following tree diagram: Tree=

<img width="500” height=" alt=" Tree=" data-cke-saved-src="https://s2.lite.msu.edu/res/msu/statlib2msu/Ashoke/Probability/Images/GasStation2.png" src="https://s2.lite.msu.edu/res/msu/statlib2msu/Ashoke/Probability/Images/GasStation2.png" tree"="Tree&quot;" diagram:="Diagram:" encrypturl="no" 400"="400&quot;" "="&quot;" tank="Tank" full="Full">

[Answer all questions below to 3 decomal places. The answers must be given as a fraction, not percentage. For instance, if the answer is 20.7% , it must be typed as 0.207.]

What is the probability that a randomly chosen customer uses diesel and filled her car's tank to the full? i.e. P(Diesel Full) =

What is the probability that a randomly chosen customer filled her car's tank to the full? i.e. P(Full) =

If a randomly selected customer filled her car's tank to the full, what is the probability that she uses diesel? i.e. P(Diesel | Full) =

Explanation / Answer

From the given information we have

P(unleaded) = 0.60, P(premium) = 0.35, P(diesel) = 0.05

And

P(Full | unleaded) = 0.70, P(Full | premium) = 0.30, P(Full | diesel) = 0.90

The probability that a randomly chosen customer uses diesel and filled her car's tank to the full is

P(Full | diesel) P(diesel) = 0.90 * 0.05 = 0.045

The probability that a randomly chosen customer filled her car's tank to the full is

P(Full) = P(Full | unleaded) P(unleaded) + P(Full | premium)P(premium) +P(Full | diesel)P(diesel)= 0.70*0.60 +0.30*0.35 +0.90*0.05 = 0.57

The probability that she uses diesel,  if a randomly selected customer filled her car's tank to the full is

P(Diesel|Full) = [P(Full | diesel)P(diesel)] / P(Full) = 0.045 / 0.57 = 0.0789

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