Solving Jack's Ticket Problem at Maryland University

[hholzer]

Problem: At a state university in Maryland, there is hardly enough space
for students to park their cars in their own lots. Jack, a student who parks
in the faculty parking lot every day, noticed that none of the last 10 tickets
he got was issued on a Monday or a Friday. Is it wise for Jack to conclude
that the campus police do not patrol the faculty parking lot on Mondays and
on Fridays? Assume that police give no tickets on weekends.

Solution: Suppose that the answer is negative and the campus police patrol
the parking lot randomly; that is, the parking lot is patrolled every day with
the same probability. Let A be the event that out of 10 tickets given on
random days, none is issued on a Monday or on a Friday. If P(A) is very small,
we can conclude that the campus police do not patrol the parking lot on these
two days. [...]

I don't follow that conclusion. Why can we conclude such if P(A) is small?
If P(A) is small, then shouldn't this be taken to mean that receiving a ticket
on one of the weekdays besides Monday or Friday is very low?

 

[jgm340]

Problem: At a state university in Maryland, there is hardly enough space
for students to park their cars in their own lots. Jack, a student who parks
in the faculty parking lot every day, noticed that none of the last 10 tickets
he got was issued on a Monday or a Friday. Is it wise for Jack to conclude
that the campus police do not patrol the faculty parking lot on Mondays and
on Fridays? Assume that police give no tickets on weekends.

Solution: Suppose that the answer is negative and the campus police patrol
the parking lot randomly; that is, the parking lot is patrolled every day with
the same probability. Let A be the event that out of 10 tickets given on
random days, none is issued on a Monday or on a Friday. If P(A) is very small,
we can conclude that the campus police do not patrol the parking lot on these
two days. [...]

I don't follow that conclusion. Why can we conclude such if P(A) is small?
If P(A) is small, then shouldn't this be taken to mean that receiving a ticket
on one of the weekdays besides Monday or Friday is very low?

If P(A) is small, then the probability of [receiving 10 tickets overall but 0 tickets on Monday or Friday] is small.

If the probability of not receiving any tickets on Monday or Friday is very low under the assumption that tickets are given out uniformly across all days, then this gives us reason to believe that our assumption is wrong. i.e., it gives us reason to believe that tickets are NOT given out uniformly across all days. P(A) is much higher (it actually equals 1) under the assumption that no tickets are ever given out on Friday or Monday, so it kind of makes sense to conclude this.

However, it's a bit of a jump to actually conclude that they don't give out tickets on Monday or Friday. There's also a very, very grave error in the interpretation given in the "solution". There is an implicit assumption that nothing else could be as surprising as not receiving any tickets on a Monday or a Friday. This is just false. Jack would have asked the question if he hadn't received any tickets on any two days of the week. There are 5 choose 2 = 10 types of outcomes that would have been just as surprising to Jack. This means we really can't conclude anything.

 

[tiny-tim]

hi hholzer!

Solution: Suppose that the answer is negative and the campus police patrol
the parking lot randomly; that is, the parking lot is patrolled every day with
the same probability. Let A be the event that out of 10 tickets given on
random days, none is issued on a Monday or on a Friday. If P(A) is very small,
we can conclude that the campus police do not patrol the parking lot on these
two days. [...]

I don't follow that conclusion. Why can we conclude such if P(A) is small?
If P(A) is small, then shouldn't this be taken to mean that receiving a ticket
on one of the weekdays besides Monday or Friday is very low?

No, if P(A) is small, that means that receiving ten tickets on one of the weekdays besides Monday or Friday is very low.

Receiving one ticket is obviously a lot higher!

eg suppose P(A) = 1%.

that means, if the tickets were random, there's only a 1% chance that all ten tickets would be Tuesday to Thursday ​

 

[jgm340]

The probability is actually (3/5)^10, which is about 0.006, which is 0.6%

3/5 is the probability that any given ticket will NOT be given on a Monday or a Friday, and there are 10 independent tickets being given.

The probability of something just as surprising happening is actually 10 times that, or 6%.

 

[hholzer]

If P(A) is small, then the probability of [receiving 10 tickets overall but 0 tickets on Monday or Friday] is small.

If the probability of not receiving any tickets on Monday or Friday is very low under the assumption that tickets are given out uniformly across all days, then this gives us reason to believe that our assumption is wrong. i.e., it gives us reason to believe that tickets are NOT given out uniformly across all days. P(A) is much higher (it actually equals 1) under the assumption that no tickets are ever given out on Friday or Monday, so it kind of makes sense to conclude this.

Thanks for your cogent reply. The key here is realizing, under the given
assumption, that if P(A) is small, then it contradicts our assumption.
Which, as you have pointed out, suggests that the police do not patrol
the parking lot randomly.