the subway train has 18 doors and starts with 15 passenger. if each passenger is equally likely to get off at any station and the passengers leave the train independently, find the probability that 2 or more passengers leave the train through the same door
YOU ARE DOOMED!!
To find the probability that 2 or more passengers leave the train through the same door, we can use the concept of complementary probability. First, let's find the probability that no passengers leave the train through the same door.
The first passenger can choose any of the 18 doors. The second passenger also has 18 doors to choose from, but they need to choose a different door than the first passenger. So, the second passenger has 17 doors to choose from.
The third passenger needs to choose a door different from the first two passengers. They have 16 doors to choose from. Similarly, the fourth passenger has 15 doors to choose from, and so on.
Therefore, the probability that no passengers leave the train through the same door is given by:
P(no passengers leave through the same door) = (18/18) * (17/18) * (16/18) * ...
Now, we need to find the complementary probability, which is the probability that 2 or more passengers leave the train through the same door. This is equal to 1 minus the probability that no passengers leave through the same door.
P(2 or more passengers leave through the same door) = 1 - P(no passengers leave through the same door)
Now, we can calculate the probabilities.
P(no passengers leave through the same door) = (18/18) * (17/18) * (16/18) * ...
P(2 or more passengers leave through the same door) = 1 - P(no passengers leave through the same door)
Please note that the calculation can be quite complex. If you want a numerical value for the probability, you can use a calculator or a programming language with a loop to approximate the result by multiplying the terms until you reach a desired level of precision.