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