Probability of meeting

thickhead · 2016-09-20 20:53:43

thickhead wrote:

mr.wong · 2016-09-21 21:51:10

Let us postpone the above problem temporarily and consider the following one . ( A diagram will be much helpful to clarify disputes ! )

Related problem ( I ) :

A boy and a girl have dated to meet at a place . They will arrive there randomly within 60 minutes . The boy is willing to wait for the girl for 30 minutes while the girl is willing to wait for the boy for 20 minutes . Find
(1) the probability of their meeting at the place .
(2) the expectation of the waiting time of the boy .
(3) the expectation of the waiting time of the girl .

bobbym · 2016-09-22 01:20:20

I for one will use the same method on this problem as the other problem. So we will have more controversy instead of less.

thickhead · 2016-09-22 16:20:13

mr.wong · 2016-09-22 20:09:57

Hi thickhead ,

I get the same probability = 47 / 72 .
But for the expectations , will you recognize that their waiting times
should differ so greatly ? ( about 3 times )

mr.wong · 2016-09-22 20:13:08

Hi bobbym ,

For the original problem if I separate the 60 minutes into 3 periods ,
I get the following results .
(I) 1st period : from 0 min. to 20 min.
Probability of meeting = 1/2 .
Boy's average waiting time = ( 10 + 30 ) / 3 = 40 / 3 min.

(II) 2nd period : from 20 min. to 40 min.
Probability of meeting = 2/3 .
Boy's average waiting time = ( 10 +10 + 10 ) / 3 = 10 min.

(III) 3rd period : from 40 min. to 60 min.
Probability of meeting = 1/2 .
Boy's average waiting time = ( 10 + 10/3 + 10/3 ) / 3 = 50 / 9 min.

Thus the average probability of meeting within the 60 min.
= (1/2 + 2/3 + 1/2 ) / 3 = 5/9 .
The expectation of the boy's waiting time within the 60 min.
= ( 40/3 + 10 + 50/9 ) / 3
= 260 / 27 mins. ( about 9.63 min. )
which is consistent with the result I got in # 8 .

Will your results also be consistent if the 60 min. also be
divided into 3 periods ?

bobbym · 2016-09-22 20:50:01

Hi;

As soon as I can I will recompute the first problem so I can be sure, then I will post a solution to it which should remove any doubts.

latest?cb=20111008200709

But for now, for your new problem I am getting

(1) 47 / 72

(2) 185 / 12

(3) 515 / 54

thickhead · 2016-09-22 22:44:51

How smart!!

Correction:
probability of meeting=47/72
Average boy's waiting time=7.135494618 minute
Average girl's waiting time=4.218827951
minute

bobbym · 2016-09-22 23:20:52

Hi mr.wong;

In post #31 you asked if I could come to the answer you have.

A boy and a girl have dated to meet at a place.
They will arrive there randomly within 60 minutes and are willing to wait for one another for 20 mins. What is the probability of their meeting at the place?
What is the Expected waiting time of the boy?

There are 4 possibilities.

1) boy comes first and girl does not come within 20 minutes, boy waits 20 minutes
2) boy comes first and girl does come within 20 minutes, boy waits g - b minutes
3) girl comes first and boy does not come within 20 minutes so boy waits 20 minutes
4) girl comes first and boy does come within 20 minutes so boy waits 0 minutes

We let g = the girls arrival time, b = the boys arrival time.

Using rules 1 to 4 we can form the following piecewise function.

Integrating this:

Unless you can find an error in my rules 1 to 4 then that is the answer.

thickhead · 2016-09-22 23:35:07

I have experimented with sample space.
Boy comes at t=30.
girl comes comes on different days at times varying from 11 to 50 minutes, each day is a sample space.

boy's arrival time 	girl's arrival time	waiting time
30	11	0
30	12	0
30	13	0
30	14	0
30	15	0
30	16	0
30	17	0
30	18	0
30	19	0
30	20	0
30	21	0
30	22	0
30	23	0
30	24	0
30	25	0
30	26	0
30	27	0
30	28	0
30	29	0
30	30	0
30	31	1
30	32	2
30	33	3
30	34	4
30	35	5
30	36	6
30	37	7
30	38	8
30	39	9
30	40	10
30	41	11
30	42	12
30	43	13
30	44	14
30	45	15
30	46	16
30	47	17
30	48	18
30	49	19
30	50	20
	total waiting time	210

average waiting time at t=30 is 210/40=5.25 minutes.
Exact answer is 5 minutes and we can approach it by taking more sample points say at the interwal of 0.1 minute.
This is more in the interwal from t=0 to t=20
and less from t=40 to t=60.

Last edited by thickhead (2016-09-23 00:08:56)

bobbym · 2016-09-22 23:40:51

Me too, and I do not get your answer.

thickhead · 2016-09-23 04:37:02

Hi bobbym and mr.wong,
I am extremely sorry for my concept was based on fruitful waiting.When I take the other part into account I get waiting time in the original problem as 260/27.

bobbym · 2016-09-23 06:28:12

I do not agree with that answer either.

I suggest that mr.wong read post #34 so that he can either support the answer given there or find the error in the 4 rules.

mr.wong · 2016-09-23 15:40:48

Hi bobbym ,

There is one point that you may had omitted .
In # 34 ( 3 ) , equivalent to my portion (A) in # 8 , the waiting time of
the boy is not fixed to be 20 minutes , in fact , it is min ( 20 , 60-b ) minutes .
For 20 < b < 40 , min ( 20, 60-b ) = 20 , which corresponds to the region A1 .
For 40 < b < 60 , min ( 20, 60-b ) = 60-b , which corresponds to the regions
A2 and A3 . The boy's waiting time will decrease gradually from 20 minutes
at time 40 minute to 0 minute at time 60 minute since the boy will certainly leave at that time . This explains why your result is a bit greater than mine .

mr.wong · 2016-09-23 15:41:54

Hi thickhead ,

Thanks for your clarification .

mr.wong · 2016-09-23 15:53:42

For related problem ( I ) I got the following result :

(1) P = 47 / 72 .
(2) Expectation of boy's waiting time = 70 / 6 min. = 11 + 2/3 min.
(3) Expectation of girl's waiting time = 445 / 54 min. = 8 + 13 / 54 min.

bobbym · 2016-09-23 17:24:41

Hi mr.wong;

Just because they arrive inside of a sixty minute window does not mean they are confined to it. That they can not stay longer than it. I asked you in post #4 about that constraint.

If the girl comes at minute 5 waits 20 minutes and leaves at minute 25 and the boy comes at minute 45 why can he not stay till minute 65? Why can he not wait his 20 minutes? If the boy comes at minute 59, he can only stay a minute?

thickhead · 2016-09-23 17:57:30

bobbym,
I would adopt your convention for uniformity. I have replaced p by W_b to reflect that it is waiting time for the boy.

I get the answer 310/27.

bobbym · 2016-09-23 18:08:59

mr.wong has a different definition than the one I used. Since he is the OP, his definition is the one that counts.

thickhead · 2016-09-23 20:08:35

As for me I played a comedy of errors.(comedy because I am not giving exam)First I neglected unsuccessful waiting by the boy.I introduced it and got 190/27. This was correct for the assumption I made.I looked back and changed a correct calculation to wrong one and got 260/27.Then I looked at bobbym's 4 rules and got 310/27. Now I read about min(20,60-b) and getting 260/27 again.

thickhead · 2016-09-23 22:15:24

For the second problem I am getting 35/3 and 445/54 for the conditions imposed.

mr.wong · 2016-09-24 00:38:23

bobbym wrote:

Hi mr.wong;
Just because they arrive inside of a sixty minute window does not mean they are confined to it. That they can not stay longer than it. I asked you in post #4 about that constraint.
If the girl comes at minute 5 waits 20 minutes and leaves at minute 25 and the boy comes at minute 45 why can he not stay till minute 65? Why can he not wait his 20 minutes? If the boy comes at minute 59, he can only stay a minute?

Hi bobbym ,

The boy will not wait any longer after minute 60 because it is
meaningless . The girl surely will not appear after that time . If
I were the boy I will leave immediately after minute 60 and go
to other place to find the girl !

bobbym · 2016-09-24 04:39:56

Hi mr.wong,

Thanks for the rule clarification, future problems should be much smoother.

mr.wong · 2016-09-24 19:56:23

Hi bobbym ,

Assumed that the boy and the girl are willing to wait for the same
time , can we find anything if a graph is drawn relating the willing
waiting time , the probability , and the expectation ?

bobbym · 2016-09-24 23:26:05

Hi;

You mean in terms of waiting time t?

Math Is Fun Forum

#26 2016-09-20 20:53:43

Re: Probability of meeting

#27 2016-09-21 21:51:10

Re: Probability of meeting

#28 2016-09-22 01:20:20

Re: Probability of meeting

#29 2016-09-22 16:20:13

Re: Probability of meeting

#30 2016-09-22 20:09:57

Re: Probability of meeting

#31 2016-09-22 20:13:08

Re: Probability of meeting

#32 2016-09-22 20:50:01

Re: Probability of meeting

#33 2016-09-22 22:44:51

Re: Probability of meeting

#34 2016-09-22 23:20:52

Re: Probability of meeting

#35 2016-09-22 23:35:07

Re: Probability of meeting

#36 2016-09-22 23:40:51

Re: Probability of meeting

#37 2016-09-23 04:37:02

Re: Probability of meeting

#38 2016-09-23 06:28:12

Re: Probability of meeting

#39 2016-09-23 15:40:48

Re: Probability of meeting

#40 2016-09-23 15:41:54

Re: Probability of meeting

#41 2016-09-23 15:53:42

Re: Probability of meeting

#42 2016-09-23 17:24:41

Re: Probability of meeting

#43 2016-09-23 17:57:30

Re: Probability of meeting

#44 2016-09-23 18:08:59

Re: Probability of meeting

#45 2016-09-23 20:08:35

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#46 2016-09-23 22:15:24

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#47 2016-09-24 00:38:23

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#48 2016-09-24 04:39:56

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