At the highway is the restaurant "Weeping Willow". The number of cars and trucks traveling along this road is 4:1. On average, 8%

Question

At the highway is the restaurant "Weeping Willow". The number of cars and trucks traveling along this road is 4:1. On average, 8% of truck drivers and 2% of car drivers stop for a meal here. What is the likelihood that the driver who comes to eat is the driver of a car?

Answer 1

Badly written question (getting a lot of those these days)

If I assume the question is as follows we might be able to come up with an answer.

"Along the highway there is a restaurant called the "Weeping Willow". The ratio of cars to trucks traveling along the highway is 4:1. On the average, 8% of truck drivers and 2% of car drivers passing by stop for a meal at the restaurant. What is the likelihood that a random driver entering the restaurant to eat is a car driver?

This is a question in conditional probability. In other words, given that a driver has stopped for a meal, what is the probability that the driver is an auto driver.

Out of 500 random vehicles passing by, 400 will be auto drivers and of those 400*.02 or 8, will stop to eat.

Out of the same 500 passing by, 100 will be truck drivers and of those 100*.08 or 8 will stop to eat.

Out of the 16 people stopping to eat (of the 500 random vehicles), half will be truck drivers and half will be car drivers. (only drivers are counted, not passengers).

The likihood ratio is then 1:1. ( ie there is a 50% chance that a driver walking thru the door of the Weeping Willow restaurant for a meal will be a car driver).

We can also write this as a conditional probability statement.

P(A|B) = [P(B|A)P(A)] / [P(B|A)P(A) + P(B|A^c)P(A^c)]

Where B is stoping for a meal

and A is being a car driver with the overall probability of being a car driver = .8

A^c is being a not car (truck) with the overall probability of .2

and we are given the probability of B|A (stopping if car driver) as .02

and the probability of B|A^c (stopping if truck driver) as .08

P(B|A^c)·P(A^c) = .08*.2 or .016. P(B|A)·P(A) = .02*.80 or .016

Thus P(A|B) = .016 / (.016+.016). = .5 , so the likelyhood of being a car driver if entering the restaurant is .5 or 50%

So logic, or laws of probability, the answer is the same.

Answer 2

Hi Berezka, we know that the ratio of trucks to cars in this scenario is 4 to 1. This means that for every 4 truck drivers, there will be one car driver that stops for a meal there. If you flip the order, it would be that for every 1 car driver that stops, there will be 4 truck drivers that stop. The question is asking about the likelihood that the driver is in a car, so that would be 1:4. Another way of writing a ratio is as a fraction. So the correct answer, which is the probability or likelihood that the driver who comes to eat is the driver of a car, is 1/4.