Coffee Time!

Have you ever plan to meet up with friends but been less than specific with your directions? Then this word problem is for you!

Ben, Colette, and Ruby agree to meet at “the coffee shop by the park.” Unfortunately, there are three coffee shops by the park, and they forget to specify which one. What’s the probability that Ben, Colette, and Ruby will all go to different coffee shops?

When I was in college, I joined the school's newspaper. A great perk was having a dedicated hang-out space at school! Better yet, I learned the 5Ws: who, what, where, when, why; of journalism. It's the 5 questions journalist are trained to ask. These are practical questions for any occasion, and in this case, we're missing "where?".

But does it always matter? Perhaps by being slightly less specific, we had the opportunity to explore new coffee shops! Let Kermit explain ridiculous optimism!

Our town of Fort Lee is becoming quite browsable. Sometimes, wandering randomly can take us out of self imposed ruts. When you're in the mood for exploring coffee shops, why not give our latest coffee house addition, Mavi's, a try?

And now, for our answer. In order to miss each other, we each had to be at different coffee shop simultaneously. Using numbers to number each of us from 1 to 3, and shops from 1 to 3, this table systematically shows all 27 possible ways that we can be in the 3 shops:

The lines (10, 11, 14, 15, 18, 19) show that there are 6 cases where we can completely miss each other. Hence, the probability of missing each other is 6 out of 27 cases = 2/9.

Coming up with all 27 combinations was a lot of exacting work. Now that we've done so, it's a little easier to understand how to compute the key cases. Let's compute all cases where we miss each other. To do so, the first person can choose 3 possible stores, the second person can choose 2 remaining choices, and the last has 1 choice. A total of 3 × 2 × 1 = 6 ways to choose stores (see highlighted lines above). Notice how we frame the selection by number of stores. It may make sense to think about increasing the number of store choices, which must then increase the number of ways we can miss each other. That is actually the commonly taught permutation without-replacement -- since we're avoiding each other -- for how n things can be selected k separate ways (n ≥ k); P(n, k) = n (n-1) (n-2) ...(n-k+1) for k factors. Now all possible ways that we can select to be in stores are 3 stores for the first of us, 3 stores for the second of us, and 3 stores for the last of us = 3 × 3 × 3 = 27. That is the permutation with-replacement -- since we want to meet; P(n, k) = n n ... n for k factors = n^k. Again, it should make sense that as the number of stores increases, there are more choices for meeting up.

Finally an easier way to compute the combined probability of us missing each other is to use the individual probability of us missing each other. It does not matter who arrives at the first store so the first person has 3/3 ways to miss everyone! Once a store is occupied, the second person has a probability of 2/3 ways to miss the first person; and the third person has a probability of 1/3 ways to miss the first 2. A combined 3/3 × 2/3 × 1/3 = 2/9 ways of missing each other.

Probability calculations can be quite difficult. Diagramming helps to understand small cases. We can use this small example to examine how more friends and/or coffee shops affects the outcome.

As for us, have no fear. With our ubiquitous phones, we quickly reconnected.

Contact:
Ruby Yao and Benedict Zoe, Mathnasium of Fort Lee
201-969-6284 (WOW-MATH), fortlee@mathnasium.com
246 Main St. #A
Fort Lee, NJ 07024

Happily serving communities of Cliffside Park, Edgewater, Fort Lee, Leonia, and Palisades Park.

More reading:
http://cns-web.bu.edu/~eric/EC500/attachments/ON(2d)LINE(20)READINGS/ballsinboxe.pdf

Photo: yelp