### Brain Teasers

# Walk in the Park

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

Alice and Bob agree to meet in the park. They agree to each show up some random time between 12:00 PM and 1:00 PM. Wait 20 minutes for the other person, or until one o'clock, whichever comes first.

Assuming they stick to their word, what is the probability that they will meet?

Assuming they stick to their word, what is the probability that they will meet?

### Hint

They are equally likely to show up at any time.However their probability of meeting will change depending on the time.

Try looking at it through only one person's perspective.

### Answer

Their probability of meeting is 5/9There are four events that can happen.

1. Bob arrives first and Alice meets him while he is waiting.

2. Bob arrives first, leaves and then Alice shows up.

3. Alice arrives first and Bob meets her while she is waiting.

4. Alice arrives first, leaves and then Bob shows up.

Also there are 3 main time segments that different things can happen during. from 12:00 12:00-12:20, 12:20-12:40, 12:40-1:00.

There is a 1/3 chance of Alice showing up in any of these times.

If Alice shows up from 12:20-12:40

Then there are 20 minutes before and after Alice's arrival that Bob could arrive and they would meet. So 40 minutes out of the hour that Bob could arrive and they will meet. (40/60)

If Alice shows up between 12:00 and 12:20 there are 20 minutes after her arrival, and all the time between 12:00 and when she arrived. This time averages out to be 10 minutes (remember we are assuming she showed up between 12:00 and 12:20) So that's 30 minutes out of the hour that Bob could show up and they would meet. (30/60)

From 12:40-1:00 you have a similar situation. There's twenty minutes before Alice's arrival that Bob could have shown up. And all the time till 1:00, which again averages out to 10 minutes. So again 30 minutes out of the hour.

(30/60)

Each of those three are equally likely with 1/3 probability so

1/3(30/60 + 40/60 + 30/60)

1/3(5/3)

5/9

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## Comments

OMG SHUT UP

Nice teaser.

i have a headache.

That teaser hurt my soul.

is an anser soposed 2 b that long

I know, but it was the only way I could think to explain it. Believeme I am open to a shorter explination

ok, i did not get that!

confusing...

WAY to complicated and purely math no fun!

Really nice teaser. A good bit of hard thought never hurt anyone. Well done!

If they have both agreed to wait until 10 then the probability is 100% that they will meet. Or what am I missing?

What the heck was that? I almost got my brains lost

Nice teaser.

hard, HARD, HARD!!!

confusing

Nice one. I think your answer explained the teaser very well.

But couldn't the waiting time have been like 12.37-12.57? Ok... well even if I completely missed that it was still HARD!!! But it was good in a way. If you have hours to do it in- or you could just do it till one o'clock, whichever comes first. Ok that was a lame attempt at a joke.

I didn't get your explanation... I guess that I should just stop reading probability teasers 'cause they make me feel dumb . Oh, well

Oh... and don't be such a know-it-all bjoel4evr!

I didn't understand any of that!

I agree with the Missileman. Deceptively confusing puzzle looks easy until you jump in.

Good teaser, boggled my brain there for a moment but I got it!

Excellent job. Well thought out!

Apr 27, 2006

This riddle has a flaw. You can't assume that they will show up at 120. 12:20 or 12:40 unless you put that in the problem itself... any random time means any random time. It is impossible to solve otherwise.

I didn't assume she showed up at any of those times. I suggested breaking the hour up into different pieces as the probability of meeting up is different during each interval.

I approached the problem slightly differently. I broke the time period into the three 20 minutes sections. There are then three relevant possibilities:

1) They arrive in the same period. They always meet in these cases.

2) They arrive in non-adjacent periods. They never meet in these cases.

3) The arrive in adjacent periods. As explained above, there is a 50% chance they meet in these cases.

There are 3 x 3 = 9 combinations of periods they can arrive in.

There are 3 combinations where they both arrive in the same period: {(1,1), (2,2), (3,3)}

There are two combinations where they arrive in non-adjacent periods: {(1,3), (3,1)}

That leaves four combinations where they arrive in adjacent periods {(1,2), (2,1), (2,3), (3,2)}.

Therefore there is a:

3/9 x 1 + 2/9 x 0 + 4/9 * .5 = 5/9

probability of meeting.

1) They arrive in the same period. They always meet in these cases.

2) They arrive in non-adjacent periods. They never meet in these cases.

3) The arrive in adjacent periods. As explained above, there is a 50% chance they meet in these cases.

There are 3 x 3 = 9 combinations of periods they can arrive in.

There are 3 combinations where they both arrive in the same period: {(1,1), (2,2), (3,3)}

There are two combinations where they arrive in non-adjacent periods: {(1,3), (3,1)}

That leaves four combinations where they arrive in adjacent periods {(1,2), (2,1), (2,3), (3,2)}.

Therefore there is a:

3/9 x 1 + 2/9 x 0 + 4/9 * .5 = 5/9

probability of meeting.

This is a standard geometric probability problem. Let the x axis be Alice's arrival time and y axis be Bob's arrival time. We're interested in the region inside a 60minute * 60minute square that corresponds to the event that they meet.

Notice this region is bounded by the line y=x shifted 20minutes up and 20 minutes down. This is because whatever time Alice arrives, Bob needs to be within 20 minutes of her time for them to meet.

The area of this region is 60 * 60 - 40 * 40. 40 * 40 is the area of the sum of the two triangles that aren't in this region. You'll see it clearly if you draw the graph.

The total area is 60 * 60. Hence (60-40)(60+40) = 2000 / 3600, so the answer is 5/9.

Notice this region is bounded by the line y=x shifted 20minutes up and 20 minutes down. This is because whatever time Alice arrives, Bob needs to be within 20 minutes of her time for them to meet.

The area of this region is 60 * 60 - 40 * 40. 40 * 40 is the area of the sum of the two triangles that aren't in this region. You'll see it clearly if you draw the graph.

The total area is 60 * 60. Hence (60-40)(60+40) = 2000 / 3600, so the answer is 5/9.

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