Probability Density Function - Will I catch the train?

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yy101

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Hi! I was having some trouble with this question:

A train arrives at a station at 7:30 am, and leaves the station between 7:35 am and 7:40 am. If you arrive at the station after 7:33 am, what is the probability that you catch the train? (Hint: Construct a probability density function which describes the distribution of probability in missing the train if you arrive at the station after time t.)
Click to expand...


My thoughts:
Not sure what the 7:30 is there for, if the probability starts at 7:33.

I didn't use a PDF, as I wasn't sure what the vertical axis would be in the PDF. Instead, I used somewhat of an 'expected value' sort of strategy.
Say you arrive between 7:33 and 7:35, then Pr(catch)=1. We know Pr(arrive between 7:33 and 7:35) = 2/7
Say you arrive between 7:35 and 7:40. Pr(catch) = 0.5, as an average - We know Pr(arrive between 7:35 and 7:40) = 5/7
So, Pr(catch)=1×27+0.5×57=914Pr(catch)=1\times \frac{2}{7} +0.5\times \frac{5}{7} = \frac{9}{14}
The answer doesn't match the official solution, though I don't really understand what went wrong. The official solution used a probability density function, which I don't understand - I think it's more of a conceptual issue. Any help or explanation would be very much appreciated!

Thank you in advance!
 
We know Pr(arrive between 7:33 and 7:35) = 2/7-- can you explain why this statement is true? Are you sure that if you arrive after 7:33 that you will arrive by 7:40? Is there a law forbidding you from arriving at 7:45. If yes, then how is this law enforced?
 
Thanks for your reply; I wasn't sure about that either, at first, but I inferred from the solution that you had to arrive at the station between 7:33-7:40. I've attached a photo of the solution below:
1633387314331.png

I don't understand what the vertical axis in the pdf is meant to be.
If I interpret it as the probability that you miss the train, then why is there 2/15 chance of missing the train when you arrive at 7:33? Also, why is, say, 7:32 included on the t-axis when you certainly don't arrive at that time?

Thank you!
 
When I look at the graph it clearly shows that t goes from 0 to infinity. You do realize that the bold print is the graph.
 
Oh...ok...I see...thanks for pointing that out.
Then if you arrive at 7:41, there's 0 chance of missing the train? I don't see how that makes sense...
 
Last edited:
The probability that you arrive at 7:33 is 0.
The probability that you arrive at 7:38 is 0.
The probability that you arrive at 7:42 is 0.
That is, the probability that you arrive at any exact time is 0.
Before moving on you need to understand what I wrote above.

If you arrive after 7:30, there is some probability that you make the train.
If you arrive after 7:31, there is some probability that you make the train.
If you arrive after 7:33, there is some probability that you make the train.
If you arrive after 7:34, there is some probability that you make the train.
The 4 probabilities above are the same. Do you see that? Call this common probability p.
Now, if you arrive after 7:35, but before 7:40, the probability that you make the train decrease to 0 as you arrive later and later.
The probability that you make the train if you arrive after 7:40 is 0.

Now the probability that you make the train if you arrive before 7:30 must be 1. That is why the area under the given curve (that matches what I have above) is 1. This happens if and only if p= 2/15.
 
yy101 said:
Oh...ok...I see...thanks for pointing that out.
Then if you arrive at 7:41, there's 0 chance of missing the train? I don't see how that makes sense...
Click to expand...
You can not compute the probability of a continuous function at an exact value.
What is the probability that someone in this world is EXACTLY 6 feet tall?
What is the probability that you pump 18.384 gallons into the fuel tank of your car?
 
Ah, ok, I see what you mean by your explanation of "after". I went wrong partly because, I think, I was interpreting each t value as an instant in time rather than the "after". Thank you.

I just had one other question:
Say the problem was a bit different. How would you calculate the probability, given you arrive between 7:33 and 7:35, that you catch the train? Would this be 1?
 
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