Killing Time WIth Recreational Math: The Birthday Paradox on Mars

If you gather 23 people into a room and compare their birthdays then there is a 50-50 chance that two people will have birthdays that match.

This seemingly small number of people is called the Birthday Paradox and it surprises people because there are 365 days in the year (ignoring leap years) yet only 23 people are needed to generate a 50% probability of a match.

In this post not only I am going to simulate the birthday paradox here on Earth using some Python code but I will go further and work it out for other planets as well.

The Code

Normally I do this sort of thing using Microsoft Excel but in this case a simple interpreted program is the way to go.

# This code released by Procrastilearner under the CC BY-SA 3.0 license.

import math
import random
import array
import time
import datetime

num_people=23 # The number of people in the group
num_trials=10000 # The number of trials to test out to generate the statistics
days_in_year=365 # The numbers of planet days in the planet's year
time1=time.time() # Get the time
random.seed(int(time1)) #Seed the random function generator

# Make the time output more readable
readabletime = datetime.datetime.fromtimestamp(time1).isoformat()
print("*************************")
print("Start program")
print("date and time = "+readabletime)

matches=0 # Count up the number of birthday matches in the group
k=0 # This is the counter for the number of trials

# Use nested loops to count up the matches
# The large loop is for each new trial (i.e. a new group of people)
# The two inner loops look for matches in birthdates
while(k<num_trials):
    a = [] #initialize the array
    # print(str(float(k/num_trials))) # Uncomment this line if you want to track progress
    for x in range(0, num_people):
        a += [random.randint(1,days_in_year)] # A birthday is just a number from 1 to 365
    foundflag = False #This is the match flag; if a match is found it will be set to true
    i = 0
    while(i<num_people):
        j = i+1
        while(j<num_people):
            if a[i]==a[j]: # Check for matching birthdates in the array
                foundflag=True #If a match is found, set the flag to true
            j = j + 1
        i = i + 1
    if foundflag==True:
        matches=matches+1 # Increment the matches counter by one
    k=k+1 #increment the trial counter and loop again

# Calculate the fraction of trials that found matching birthdates
fraction=float(matches)/num_trials 

print("#trials = "+str(k))    
print("number of people = "+str(num_people))
print("days in year = "+str(days_in_year))
print("matches = "+str(matches))    
print("match fraction = "+str(fraction))    
print("Finish program")
print("*************************")

Description of the Code

Initialization

In the initialization section of the code you set up the number of people in the group ('num_people'), the number of days in the year ('days_in_year') and the number of trials you want to run ('num_trials').

For Earth, the number of people to get a roughly 50% probability of a match is 23, the number of days in the year is 365 and to get good statistics you will want to set the number of trials to something large like 1000 or more.

The Set Up

The code sets up the random function using a seed based on the time you execute the code. It initializes an array 'a' with random numbers between 1 and the number of days in the year.

The Main Part Of The Code

The heart of the code are the nested loops. The large loop controls each trial, it creates a new group of people each loop and resets 'foundflag' back to FALSE.

The inner loops just compare the numbers stored in the array looking for matches. If a match is found then 'foundflag' is set to TRUE and the match_counter is incremented later on.

Code Validation

Code Output

Wikipedia has a page dedicated to this problem and it provides answers for various group sizes using its own probability calculations. Using this code, I changed the number of people in the group size and got the following probabilities from my code.

Number of People	Wikipedia Answer	Python Code Answer
5	2.7%	2.5%
10	11.7%	11.5%
20	41.1%	42.0%
23	50.7%	50.6%
30	70.6%	70.7%
40	89.1%	88.9%
50	97.0%	97.2%
60	99.4%	99.4%
70	99.9%	99.92%
75	99.97%	99.98%

So my code seems to be behaving and it is predicting the same answers that are published in the Wikipedia article. This gives us confidence to move on to the other planets.

The Birthday Paradox on Mars

The length of the Martian year is 687 Earth days and its rotational period is 1.026 Earth days long. This means that there are 669 Mars days in a Mars year.

Plugging in 669 for 'days_in_year' and trying different numbers for 'num_people' I find that it only takes 31 people to get 50% chance for a birthday match on Mars.

The Birthday Paradox on Europa

Europa is a moon that orbits Jupiter. Jupiter orbits the Sun every 4,333 Earth days. Europa is tidally locked with Jupiter (one side always faces Jupiter) and it orbits its mother planet every 3.55 Earth days. This means that there are 1220 Europa days in a Europa year.

Plugging the number 1220 into 'days_in_year' and testing out different values for 'num_people' I find that it only takes 41 people to get a 50% chance of matching birthdays on Europa.

There Is No Birthday Paradox on Jupiter

Jupiter orbits the Sun every 4,333 Earth days and its days is slightly less than 10 Earth hours long. This means that there are 10,476 Jupiter days in a Jupiter year. Plugging in the relevant numbers into the code we find that it takes 122 people to reach a 50% chance of matching birthdays for Jovian inhabitants.

This fairly large number means that it would be rare to find matching birthdays in any group and the birthday paradox would really not exist on this gas giant.

The Birthday Paradox on "Pluto Is A Planet"

Pluto orbits the Sun every 90,560 Earth days (248 Earth years) and its day is 366.7 Earth days long. This means that there are 247 Pluto days in a Pluto year. Plugging in the numbers into the code and we find that it only takes a group of 19 people to get the 50% birthday paradox result on Pluto.

The Birthday paradox is even stronger on Pluto than it is on Earth. Also your birthday would be 367 Earth days long. This is why Pluto is such a kick-ass planet.

Closing Words

The birthday paradox is an interesting example of how probabilities can surprise you. From a seemingly large set of 365 days in a year, it only takes 23 people to achieve a 50% chance of a match-up.

On some of the other planets and moons in our solar system the paradox still holds but on a few planets like Jupiter it kind of disappears.

Thank you for reading my post.

Other Posts In My Recreational Math Series

1. Testing the excel random function.
2. Make Your Own Bell Curve
3. Strange Attractors.
4. Let's Travel To Alpha Centauri
5. Calculate Sunset and Sunrise Times
6. Conway's Game of Life.
7. Caffeine Half-Life.
8. Let's Simulate a Radioactive Sample
9. Journey Through the Centre of Psyche
10. Star Trek Into Darkness

Post Sources

[1] Wikipedia: Birthday Problem
[2] Better Explained: Understanding the Birthday Paradox
[3] Scientific American: Probability and the Birthday Paradox
[4] Python 3.6.5
[5] Mars
[6] Europa
[7] Jupiter
[8] Pluto