Question

Two people agree to meet on January 9, 2005 between 6:00 P.M. and 7:00 P.M., with the understanding that each will wait no longer than 20 minutes for the other. What is the probability that they will meet?

• A. \(\dfrac{5}{9}\)
• B. \(\dfrac{7}{9}\)
• C. \(\dfrac{2}{9}\)
• D. \(\dfrac{4}{9}\)

Hint:

Model both arrival times as points in a 60 by 60 square and find the area where the gap between them is at most 20 minutes.

Answer 1

The Correct Option is A

Step 1: Set up the sample space.
Let \(x\) be the number of minutes after 6:00 P.M. that the first person arrives, and \(y\) be the number of minutes after 6:00 P.M. that the second person arrives. Both \(x\) and \(y\) can be any value from \(0\) to \(60\), and since arrival time is random within the hour, every point \((x,y)\) in the \(60\times 60\) square is equally likely.

Step 2: Write the meeting condition.
The two people actually meet only if neither has already left by the time the other arrives, that is, if the gap between their arrival times is at most 20 minutes:
\[ |x-y|\le 20 \]

Step 3: Find the area where they do NOT meet.
It is easier to first find the region where \(|x-y|>20\), meaning one person arrived and left more than 20 minutes before the other showed up. This condition splits into two triangular regions in the square: one where \(x-y>20\) and one where \(y-x>20\). For \(x-y>20\), \(x\) ranges from \(20\) to \(60\) (a length of \(40\)) and for each such \(x\), \(y\) ranges from \(0\) to \(x-20\), forming a triangle with both legs equal to \(40\). Its area is
\[ \frac{1}{2}(40)(40) = 800 \]
By symmetry the triangle for \(y-x>20\) also has area \(800\). So the total do not meet area is
\[ 800+800 = 1600 \]

Step 4: Find the area where they DO meet.
The full square has area \(60\times 60 = 3600\). Subtracting the non-meeting area:
\[ 3600 - 1600 = 2000 \]

Step 5: Convert area to probability.
Since every point in the square is equally likely, the probability of meeting is the meeting area divided by the total area:
\[ P(\text{meet}) = \frac{2000}{3600} = \frac{5}{9} \]

Final Answer:
The probability that the two people meet is \(\dfrac{5}{9}\). \[ \boxed{\dfrac{5}{9}} \]