The Probability of True Love

1
The Probability of True Love

• A Problem in Discrete Probability With a Little
  Help From Calculus

2
Rules of the Dating Game

• It is generally considered risky 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 him or her
  or say no, and move on to another person.
• Once someone has been passed over, you cannot go
  back. No is forever.
• If there are N people possible to date before the
  time you die, how can you maximize the
  probability that you select your best match?

3
The Play-the-Field Strategy

• Date K people without making a selection.
• Continue to date after the Kth person and
  select the first person better than any of the
  first K people.
• Its true love baby

4
The Max-Min Dynamic

• If N is the total number of people you might date
  in a life time, then, true-love is one of these
  N choices.
• If you choose K to be too small, you risk not
  meeting your best mate.
• If you choose K to be too big, you risk passing
  your best mate by in search of someone better.

5
A Probability Model to Maximize the
Play-the-Field Strategy

• The Goal Find the value of K (relative to N)
  that gives the greatest probability of selecting
  the best match from N choices.
• Develop a function P(K) that will compute the
  probability of success as a function of K.
• The domain of P(K) is K0,1,2,,(N-1)
• If K0, then you will marry the first person you
  date. If KN-1, then you will marry the last
  person you might meet before you die.

6
The Skeleton
P(best is in position n) P(selecting best Given
best is in position n))
where p1 is easy to figure out
but p2 takes some more thought.
7
Building P(K) contd
The probability of the best person being selected
if he or she is one of the first K is
, for 1 to K. Thus we get, for the
first K positions of n. The probability of
selecting the (K1)th person given he or she is
the best is . Thus we now have,
for the first K1 positions of n.
8
P(K) contd The K2 position
P(selecting K2 GIVEN K2 is the best) One
way to interpret this probability is the
probability that the best person dated out of the
K1 people is among one of the first K. (This
best person among the first K can be thought of
as the overall 2nd best, because we are now
assuming that we will find someone better and
stop the process)
9
P(K) contd The K3 position
Using the previous argument, the probability of
selecting K3, given K3 is the best is
equivalent to finding the probability that of the
K2 people dated, the best so far is among the
first K. Thus, Continuing this argument for
K4 all the way up to the N-1 position, we get
the following for P(K).
10
P(K), And Some Simplifying
1 2 Note The domain in 1 is
K0,1,2,,N-1. And the domain in 2 is
K1,2,,N-1
11
An Example of a Math Geek

• Lets say for some math geek, N5.
• The geek should either set K1,2,3, or 4
• Which K gives the best probability?

12
A Player

• What if this math geek is a player and N1000
• How can we find K so that his probability of
  choosing the best mate is at a maximum without
  having to calculate all the probabilities.
• We have two choices
• Write a computer program (see hand out)
• Get an assist from Calculus

13
An Assist From Calculus

• Recall,
• The sum is an approximation to the area
  underneath the function y1/x from K to N
• Calculus students will recognize this to be
  equivalent to

14
Transforming P(K)

• We can now get a function that approximates P(K)
  by substituting ln(N/K) for the sum.
• Letting we get,
• So Calculus students, how should we find this
  functions maximum?

15
The Approximate Maximum

• To find the maximum we set the derivative equal
  to zero, and solve for x.

16
Thus,.

• From we get,
• Or

17
Conclusion

• So to choose K so that we have the best chance of
  meeting true love, we simply predict the total
  number of people we might date and divide by e!
• This approximation is fairly accurate for Ngt15