Mahjong - probability

[timebomb]

Good puzzle, Jacian. It's very interesting.

Let's see. The first passenger has a 1/100 chance of sitting on his assigned seat. It is a very low chance. So the next passenger is likely to find his own assigned seat vacant. The probability of that is 99/100. I don't think it's 98/99 but I could be wrong. Anyway, if the probability of finding his own assigned seat vacant is so high for the 2nd passenger, it would mean all subsequent passengers are likely to find the same thing too.

So, my final answer is: The probability of the last passenger, Tom finding his own seat vacant is 99/100.

That was the post I wanted to send but just before I clicked on "submit reply", another thought struck me.

But what about the second last passenger, the one before Tom? He will find 2 empty seats, one of which is likely to belong to Tom but what about the other? Aha!! The other seat is likely to belong to the first passenger. So the second last passenger is faced with 2 seats, the likelihood of which is that both are not his. He will therefore randomly sit on one as the puzzle has stated. Therefore,

My final final answer is: The probability of the last passenger, Tom finding his own seat vacant is 1/2. Or a figure very close to that number

Loh K L

[jacian]

Nice guess, the answer is 0.5. You will need to use induction to get the answer.

Assuming if P1 is supposed to sit in seat 1, P2 for seat 2

For 2 seats, 2 possible scenario with each probability of 1/2
Scenario 1 - P1 gets seat 1, thus Tom gets his assigned seat with probability 1
Scenario 2 - P1 gets seat 2, thus Tom gets his assigned seat with probability 0

Total probability = (1/2)(1) + (1/2)(0) = 1/2

For 3 seats, 3 possible scenario with each probability of 1/3
Scenario 1 - P1 gets seat 1, thus Tom gets his assigned seat with probability 1
Scenario 2 - P1 gets seat 2, so P2 opts for seat 1 or 3 thus the 2 seats case with probability of 1/2
Scenario 3 - P1 gets seat 3, thus Tom gets his assigned seat with probability 0.

Total probability = (1/3)(1) + (1/3)(1/2) + (1/3)(0) = 1/2

For 4 seats, 4 possible scenario with each probability of 1/4
Scenario 1 - P1 gets seat 1, thus Tom gets his assigned seat with probability 1
Scenario 2 - P1 gets seat 2, so P2 opts for seat 1, 3 or 4 thus the 3 seats case with probability of 1/2
Scenario 3 - P1 gets seat 3, so P3 opts for seat 1 or 4 thus the 2 seats case with probability of 1/2
Scenario 4 - P1 gets seat 4, thus Tom gets his assigned seat with probability 0.

Total probability = (1/4)(1) + (1/4)(1/2) + (1/4)(1/2) + (1/4)(0) = 1/2

For 5 seats, 5 possible scenario with each probability of 1/5
Scenario 1 - P1 gets seat 1, thus Tom gets his assigned seat with probability 1
Scenario 2 - P1 gets seat 2, so P2 opts for seat 1, 3, 4 or 5 thus the 4 seats case with probability of 1/2
Scenario 3 - P1 gets seat 3, so P3 opts for seat 1, 4 or 5 thus the 3 seats case with probability of 1/2
Scenario 4 - P1 gets seat 4, so P4 opts for seat 1, or 5 thus the 2 seats case with probability of 1/2
Scenario 5 - P1 gets seat 5, thus Tom gets his assigned seat with probability 0.

Total probability = (1/5)(1) + (1/5)(1/2) + (1/5)(1/2) + (1/5)(1/2) + (1/5)(0) = 1/2

I guess the pattern is obvious for now.

[timebomb]

So I'm right but my logic is wrong You can't blame me; I've never heard of "induction".

Loh K L

[timebomb]

For my perspective of this ongoing debate: I do hope you can go back to the previous scenario/story I posed on the 3 prisoner problem and prove by contradiction why it is not half. You have yet to do that and continue to insist I am wrong. Like I say, I will be more than happy to acknowledge that you are right if you can disprove that scenario.

Sorry, Checkerboard, I didn't answer your question earlier but I was distracted by the puzzle that Jacian posted. Anyway, you have to tell me exactly which is the scenario/story you are referring to. There are so many in this thread I really don't know which is the one. But before I prove that it is not half as you have thought, I have to ask you something first.

Do you understand the 3 door puzzle? Do you agree that when the contestant switches doors, he will improve his chances of winning the car from 1/3 to 2/3?

If you answer "yes" to both questions, then I will try and explain why you are wrong with the 3 prisoner puzzle. If you say "no", then I won't bother. This is because if you don't understand the 3 door puzzle which is easier to understand, I don't want to won't waste my breath explaining the 3 prisoner puzzle to you.

Loh K L