Train and seat ticket Probability question!

Question

You are looking for a first-class ticket on a particular train. There are only two seats available in

the whole train with 7 compartments and 100 seats per compartment. Only the first compartment

is the first class. Find the probability that there is a ticket available in the first class?

Answer 1

This is 1 - P(both seats available are in the other 6 cars). There are 700 seats, and 600 seats in the other cars.

1 - C(600,2)/C(700,2) = 1 - (600*599)/(700*699) = 433/1631 = .2655

You can also calculate this using 1 - HYPGEOM.DIST(0,100,2,700,0)

Interesting bounds on this probability: it is less than 2/7, as this is the expected number of seats available in the first car. It is greater but very close to 1 - (6/7)^2 = 13/49 = .2653, which is the number we get when having two independent samples.