Once upon a time, there were two railway trains competing for the passenger traffic of 1000 people leaving from Chicago at the same hour and going to Los Angeles. Assume that passengers are equally likely to choose each train. How many seats must a train have to assure a probability of .99 or better of having a seat for each passenger?

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

Solution: A

By the central limit theorem, the number of passengers for either train is approximately normally distributed with mean 1000*1/2 = 500 and variance 1000 * 1/2 *1/2 = 250. Therefore the minimum number of seats needed equals the 99th percentile of a normal distribution with mean 500 and variance 250. The 99th percentile of a standard normal is 2.326 which implies that the 99th percentile for the number of passengers for each train is

500 + 2.326 * 2501/2 =  536.78

Hence the answer is 537.