Problem of the Week Problem B and Solution Traffic Predictions

Problem

Petr was standing at the bus stop during rush hour and started counting the passing vehicles. In the first five minutes he waited, he counted $$20$$ cars, $$25$$ vans and $$15$$ trucks.

1. Based on Petr’s sample data, what is the theoretical probability that the next vehicle will be a truck?

2. Petr counted vehicles for another five minutes and discovered that the experimental probability of a vehicle being a car was the same for his first and second samples. If Petr counted a total of $$84$$ vehicles in his second sample, how many of those vehicles were cars?

Solution

1. Petr’s sample had a total of $$20+25+15=60$$ vehicles. Since $$15$$ of these were trucks, the theoretical probability that the next vehicle will be a truck is $$\frac{15}{60}=\frac{1}{4}$$. Notice that the probability is equal to the fraction of trucks in the sample.

2. Petr’s first sample included $$20$$ cars which is $$\frac{20}{60}=\frac{1}{3}$$ of the vehicles. Thus, the experimental probability of a vehicle in the first sample being a car is $$\frac{1}{3}$$. If this experimental probability is the same for the second sample, then $$\frac{1}{3}$$ of the cars in the second sample must have been cars. Since his second sample had a total of $$84$$ vehicles, and $$\frac{1}{3}$$ of $$84$$ is $$\frac{1}{3}\times 84=28$$, it follows that $$28$$ of the vehicles in the second sample were cars.

Note: You cannot predict the individual numbers of vans or trucks in the second sample, because you don’t know the experimental probabilities of a vehicle being a van or a truck for the second sample.

Extension: If Petr had determined that the probability of a vehicle being a car was the same for his first and second samples, would it have been possible for him to have observed $$85$$ vehicles in the second sample?