The Birthday Paradox
Birthday Paradox Objectives:
STUDENTS WILL:
1. Memorize the definition for probability and for
compound events.
2. Discuss the probability of isolated and compound
events.
3. Apply knowledge of probability to find probability of
remaining event(s).
4. Differentiate between isolated and compound
events in the Birthday Paradox.
5. Propose and defend methods for solving the
Birthday Paradox using knowledge of probability.
6. Develop pseudo code for solving the Birthday
Paradox in Matlab.
Birthday Paradox

How likely is it that two people in this
room have the same birthday?

How many people are needed to ensure
that the probability of having at least two
people with the same birthday is 50%?
Birthday Paradox
Probability – the likelihood that something will
occur or be true
P(E ) 
Number of Elements
in ' E'
Size of the Sample Space
The sum of all probabilities for all events in the
sample space must sum to 1.
P ( E 1 )  P ( E 2 )  ...  P ( E N )  1 . 0
Birthday Paradox
Compound Events – 2 or more independent
events occurring in sequence.
P ( E 1 and E 2 )  P ( E 1 ) * P ( E 2 )
Birthday Paradox
How likely is it that two people in this room
have the same birthday?
It’s actually easier to get at the inverse. What is
the probability that two people will NOT have
the same birthday?
P(E ) 
# Days/Yr
- Birthday
# Days/Yr
P(E ) 
364
365
 0 . 997
Birthday Paradox
Now that we know the probability that two
people chosen at random WON’T have the
same birthday, we can figure out the probability
that they WILL have the same birthday…
Sum of All P(E)  1 . 0
P ( match )  P ( no match )  1 . 0
P ( match )  1 . 0  P ( no match )  0 . 003
Birthday Paradox

How many people are needed to ensure
that the probability of having at least two
people with the same birthday is 50%?

Let’s see how this plays out in Excel…
# of Students # of Non-matching P(no match) P(no match) for P(match) for
Checked
Birthdays
for Nth student first N students first N students
1
365
1.000
1.000
0.000
2
364
0.997
0.997
0.003
3
363
0.995
0.992
0.008
4
362
0.992
0.984
0.016
5
361
0.989
0.973
0.027
6
360
0.986
0.960
0.040
7
359
0.984
0.944
0.056
8
358
0.981
0.926
0.074
9
357
0.978
0.905
0.095
10
356
0.975
0.883
0.117
11
355
0.973
0.859
0.141
12
354
0.970
0.833
0.167
13
353
0.967
0.806
0.194
14
352
0.964
0.777
0.223
15
351
0.962
0.747
0.253
16
350
0.959
0.716
0.284
17
349
0.956
0.685
0.315
18
348
0.953
0.653
0.347
19
347
0.951
0.621
0.379
20
346
0.948
0.589
0.411
21
345
0.945
0.556
0.444
22
344
0.942
0.524
0.476
23
343
0.940
0.493
0.507
24
342
0.937
0.462
0.538
Birthday Paradox
Now let’s try programming the Birthday Paradox
in Matlab so that we can practice our
programming skills…
Useful Matlab Functions:
sort, find, isempty, ceil, rand, sum, for, while
Birthday Paradox
function [ P ]=BirthdayParadox( N, M )
%N is the number of iterations
%M is the number of people being tested; by default, M = 23.
%1. Create random birthdates for M students.
%2. Sort the birthdates in ascending order.
%3. Determine if there are matching birthdays.
%4. If there are matching birthdays increment the MATCH counter.
%5. Repeat steps 1-4 for N iterations.
%6. Sum up all of the iterations with matching birthdays and divide by
the total number of birthdays.
end