Where am I going wrong on this probability question? A person has to travel from A to D changing buses at stops B and C en-route. The maximum waiting time at either stop can be 8 minutes each, but any time of waiting up to 8 minutes is equally likely at both places. He can afford up to 13 minutes of total waiting time, if he is to arrive at D on time. What is the probability that the person will arrive late at D?
The options given are: a) 8/13 b) 13/64 c) 119/128 d) 9/128
My logic:
Since the person can wait for max 8 minutes at each stop so he/she will be late if he/she has to wait for either 14, 15 or 16 minutes.
The number of ways this can happen is:
Person waits for 8 minutes at B:
Probability for this to occur is 1/8 (since it says that any time of waiting upto 8 minutes is equally likely at both places).
So in order to get late he/she will have to wait for either 6 or 7 or 8 minutes at C. The probability for which will be 3/8.
Person waits for 7 minutes at B:
Probability will be 1/8
So probability at C will be 2/8 (either waits for 7 or 8 min)
Person waits for 6 minutes at B:
Probability will be 1/8.
Probability at C will be 1/8 (can wait for only 8 min in order to get late).
So total probability is: (1/8)(3/8)+(1/8)(2/8)+(1/8)*(1/8) = 6/64.
Now we can consider this same scenario when he/she waits for 8 min at C.
So complete probability is 2*(6/64) = 12/64 = 3/16
And that is nowhere in the options!
Please tell me where am I going wrong.
NOTE: Here I have assumed that he he/she has to wait for atleast 1 min...if I dont assume that and also take into consideration that he/she might not have to wait at all then instead of 1/8 we will have 1/9. But that also is not giving me the correct solution.

You are assuming that the waiting time is a natural number.  A more reasonable assumption is that the waiting time is uniformly distributed over the interval $[0,8]$  You can think of the two waiting times as choosing a point in the square $[0,8] \times [0,8]$ and the chance of exceeding a total of $13$ minutes is the part of the area above $x+y=13$.  That will get you one of the choices.
