Stage 2 Mathematical Methods
Investigation: Modelling of Data
The Birthday Problem
Suppose you flip a coin and bet that it will come up tails.
Since you are equally likely to get heads or tails,
the probability of tails is 50%. This means that if you try this bet often, you should
win about half the time.
What if somebody offered to bet that at least two people in
your maths class had the same birthday. Would you take the bet?
This question is more complicated than flipping a coin, because the chance of finding
two people with the same birthday depends on the number of people you ask. If there
were only one other person in your math class, you might be surprised to find out that
she had the same birthday as you.
If there at least two people with the same birthday in an assembly of 400 people,
would you still be surprised?
Your task is to find how large must a group be so that the probability of finding
at least two people with the same birthday is at least 50%?
Let's forget about leap years when we solve this problem (no February 29 birthdays!
Sorry Madame Jarema). This way, we can assume that a year is always 365 days long.
Also, let's assume that a person has an equal chance of being born on any day of the
year, even though some birthdays may be slightly more likely than others. That will
simplify the maths without changing the result significantly.
How to go about discovering the answer to the Birthday Problem
Experimental Method
We could start by finding out the probability that in a group of two people both
people have the same birthday. To do this we would need to select several groups of
two people at random and record their birthday and then state the probability that both
share the same birthday.
We could then repeat the process by selecting several groups of three people at
random and record their birthdays and find the probability that in a group of 3 people,
at least two people have the same birthday.
Repeating this we could then select several groups of four people at random and
record their birthdays and find the probability that in a group of 4 people, at least two
people have the same birthday.
This is time consuming so to repeat this process and simulate this process more
efficiently use the website www.mste.uiuc.edu/reese/birthday/intro.html
When you have collected your data and graphed it, use the modelling process to find
the equation that best models the graph that you have drawn.
From this you should be able to find out how large a class must be to make the
probability of finding at least two people with the same birthday at least 50%?
Theoretical Method
This method is based on the theory that the probability that at least 2 people in the
group have the same birthday is equal to 1- the probability that no one in the group
has the same birthday.
The probability that both people in a group of 2 have the same birthday
=1-probability that neither person has the same birthday
=1-p(first person can have any birthday x second person can not have the same birthday)
=1-(365/365 x364/365)
=
The probability that 2 people in a group of 3 have the same birthday
=1-probability that neither person has the same birthday
=1-(first person can have any date x second person can not have the same date as the
first person x third person can not have the same date as the other two people)
=1-(365/365 x364/365x 363/365)
=
Repeat this process to generate points to graph.
Find the equation of the curve that best models this data and use it to find the number
of people in a group needed to have a 50% chance that at least two people in the
group have the same birthday.
Compare your answers from the experimental and theoretical methods.
Use calculus to confirm your answers.
Logistic Functions
Investigate the infection points of all logistic functions of the form
y=
A
where t ³ 0 and a,B and c are constants,
1+ Be-ct
by finding algebraically the value of t and y at the point of inflection.
Use this result to verify your results to the birthday problem.