2 persons are to meet between 8pm & 9pm.Each wil wait for maximum 20mins for other.Find probability of meeting

Answer 1

I assume you mean that each person will arrive at a random time between 8pm and 9pm and then wait for 20 minutes.

If the first person to show up (whichever person that happens to be) gets there after 8:40 then it's guaranteed that they will meet, because the second person has to show up no later than 9:00. What is the probability that this happens? It's the same as asking what's the probability that neither will show up before 8:40: 1/3 x 1/3 = 1/9.

If the earliest arriving person shows up before 8:40 (which will happen 8 times out of 9) the odds are 50-50.

So, the total probability is (1/9)x100% + (8/9)x50% = 1/9 + 4/9 = 5/9.

I verified this answer with a simple computer program to simulate 1 million random encounters.
10 m=0
20 randomize
30 for n=1 to 1000000
40 a=8+rnd
50 b=8+rnd
60 if abs(a-b)<1/3 then m=m+1
70 next n
80 print m/1000000
The output I got was .555529, which is almost exactly 5/9.

EDIT: I thought of another way to approach the problem. Let's look at it from the point of view of person A and see how the probability of meeting person B depends on what time A arrived.

FIRST SECTION If A arrives exactly at 8pm, what's the probability that B will show up in the next 20 minutes? That's easy, it's 1/3. And what if A arrives at, say, 8:07, then what? Well, there's a 7/60 chance that B has already arrived, in which case they definitely meet, and there's a 53/60 chance that B hasn't arrived yet, in which case the probability is 20/53 that B will arrive in the next 20 minutes. Putting this together, the probability of a meetup when A arrives at 8:07 is 7/60+(53/60)(20/53)=7/60+20/60=27/60. In general, if A arrives at x minutes past 8 (for 0
SECOND SECTION If A arrives between 8:20 and 8:40, then what? Say, for example, A arrives at 8:33. There is a 33/60 probability that B has already arrived, and if that's the case, there's a 20/33 probability that it was in the previous 20 minutes. There is also a 27/60 probability that B hasn't arrived yet, in which case there's a 20/27 probability that B will arrive in the next 20 minutes. So the probability of a meetup is (33/60)(20/33)+(27/60)(20/27)=40/60. For any value of x with 20
THIRD SECTION If A arrives at, say, 8:52, there is a 52/60 probability that B has already arrived, in which case there's a 20/52 probability that it was in the previous 20 minutes, otherwise it's 8/60 probability that B will arrive in the next 8 minutes. For any value of x with 40
Putting these three sections together, we have a piecewise defined function P(x) which represents the conditional probability of B meeting up with A assuming A arrived at x minutes past 8. P(x) starts off at 1/3, rises linearly to 2/3 when x is 20, stays flat at 2/3 until x is 40, and drops linearly to 1/3 when x is 60. The integral of this curve is the cumulative probability of B meeting up with A. We don't need calculus to find the area, a little bit of grade school geometry will suffice. The first section (a trapezoid) has area (1/3)(1/3+2/3)/2, the second section (a rectangle) has area (1/3)(2/3) and the third section (another trapezoid) has area (1/3)(1/3+2/3)/2, hence the total is 1/6+2/9+1/6=5/9

Answer 2

This problem can be broken down into 3 segments.

1 - A arrives in the "core" time between 8:20 and 8:40 where there is a 40 minute meeting time overlap. In other words, B can arrive within 40 minutes of A and meet him. The probability of this happening is 1/3 X 2/3 = 2/9.

2 - The 1st extremity. The time from 8:00 to 8 20 If A arrives at 8:00, there is only a 20 minute overlap. If he arrives at 8:20, there is a 40 minute overlap. The average is a 30 minute overlap compared to a 40 minute overlap in case 1, so that is 3/4 of the probability in case 1, or 1.5/9.

3 - Same logic as 2, but at the other time extremity, i.e. 8:40 to 9:00, giving a probability of 1.5/9

Total probability = 2/9 + 1.5/9 + 1.5/9 = 5/9

Answer 3

This is solved by looking at the difference between two uniform random variables. Let X be the first person and Let Y be the second person.

Let X ~ Uniform(0, 1)
Let Y ~ Uniform(0, 1)

then X - Y ~ f(d)

where
f(d) = 1 + d for -1 ≤ d < 0
f(d) = 1 - d for 0 < d ≤ 1

the probability

P( |X - Y| < t ) =

t
∫ f(u) du
-t

in this case we are looking for:

1/3
∫ f(u) du
-1/3

= 5/18 + 5/18 = 10/18 = 5/9

Answer 4

If person A arrives at 8:20, he will meet person B if person B arrives anywhere between 8:00 and 8:40. Thus the probability they will meet is 2/3