How to Find a SpouseA Problem in Discrete Mathematics

With an Assist From Calculus

The Rules of the Game

• It is generally considered bad form to date seriously two different people simultaneously, so you consider each person one at a time.

• You can date someone for any length of time, but eventually, you must either “select” them or say “no”and move on to another candidate.

• Once someone has been passed over, you cannot go back. No is forever.

If there are N candidates, how can you maximize the probability that you select your best match?

Alternative Interpretations• Choosing largest number in a pile

• Hitting a slow-pitch softball

What’s a good strategy?

The “Take the First Pitch”

Strategy

Fix N

A Mathematical Model

1

prob of being in Position prob of being selected given they are in PositionN

n

P k n n

Our Probability Function

Can Calculus Help?

Calculator Program

But, Calculus Can Help

1 1 1 1

1 2 1

N

k

dx

k k k N x

Differentiable Approximation1 1 1 1

1 2 1

N

k

dx

k k k N x

Define a Continuous Variable

What is Our Probability of Success?

The Moral?