Using a spreadsheet to extend our thinking.
In a group of n people, what is the probability that
at least two people have the same birthday?
Before we begin calculating the actual mathematical probabilities, we should predict what results we
expect to receive. At this point your prediction is just your best guess.
Fill in the column labeled “1st predicted probability” in the table below.
(Note: you may write your probabilities as percents.)
Number of people in the group
1st predicted probability
2nd predicted probability
Actual probability
10
20
30
50
365
Number of people needed to
have a probability of .50 (50%)
Number of people needed to
have a probability of .95 (95%)
Class notes on the complement of an event.
For this problem, it is easier to calculate the probability of the complement. (for brownie points, ask “why?”)
If our event is “The probability that at least two people share the same birthday.”, state the complement of this event in the
box below.
Class notes on the multiplication rule of probabilities.
Step 1:
Planning our Attack
Before we begin entering information into the computer, we need to plan out our strategy on paper.
Let’s start with a group of three people (n=3).
How many possibilities are there for the first person’s birthday?
Out of how many possible days in the year?
Now that you’ve selected a birthday for the first person,
how many choices do you have for the second person’s
birthday if it must be different from the first?
This fraction represents the probability
that the second person has a different
birthday from the one preceding.
Out of how many possible days in the year?
Now that you’ve selected a birthday for the first two people,
how many choices do you have for the third person’s birthday
if it must be different from the previous two?
This fraction represents the probability
that the third person has a different
birthday from those preceding.
Out of how many possible days in the year?
Using this information and the multiplication rule, find the probability that none of these three share a birthday.


=
Now that you’ve made sense of the problem, we will use a spreadsheet to find the probabilities for a
group of n people.
Step 2:
The Spreadsheet
Click on the START menu in the bottom left corner of your screen. Select PROGRAMS, then
MICROSOFT EXCEL.
If you have trouble entering
This will open the spreadsheet program.
Under the FILE menu select OPEN.
Double-Click on the file named “BIRTHDAY PROBLEM.”
formulas, copying and pasting, or
just have a question, please ASK.
In the column labeled “Number of people in group (n)” use cells A3-A367 for the numbers 1-365. To
do this type a 1 in cell A3, then click on cell A4 and enter the formula =A3+1. COPY cell A4 and
PASTE it to all cells down to A367.
In the columns labeled “Number of choices for the nth person's birthday” and “Possible days in the
year”, enter in the information we came up with earlier on paper.
In cell D3 under the column labeled “Probability that the nth person has a different birthday from those
preceding” write a formula to calculate this probability.
Remember: a probability of
*Remember, to enter a formula, you must first type an equals sign (=).
1 means a 100% chance !
COPY this formula from cell D3 and paste it into cells D4-D5.
In cell E3 under the heading “Probability that none share a birthday”, type 1. Since this cell
corresponds to a group with only 1 member, we are guaranteed that nobody else will share this
birthday!
Now, in cells E4 and E5 we are going to write a formula to calculate the “Probability that none share a
birthday” for a group of 2 and a group of 3 people.
In cell E4 type the following formula =E3*D4.
In cell E5 type the following formula =E4*D5.
Does the answer in cell E5 match up with your prior calculation on the paper?
To make some sense of this formula, think back to our work on paper. We used the multiplication rule
to calculate this probability for a group of three in the following way:
365

365
364

365
Probability that none
share a birthday in a
group of 2.
363
365
probability that the 3rd
person has a different
birthday from those
preceding
If we extend this understanding to a group of 4…
365
365

364
365

Probability that none
share a birthday in a
group of 3.
And so on for 5,6,7, …
363
365

362
365
probability that the 4th
person has a different
birthday from those
preceding
Now that we have found the probability that none share a birthday for groups of 1,2, and 3 members,
use your knowledge of the complement to write a formula for the probability that at least 2 share a
birthday (in cells F3, F4, and F5).
Now that the table is complete for n=1,2,3 we will use the POWER OF TECHNOLOGY to apply
these ideas to groups with n = 4 to 10 members. To do this, COPY cells B5-F5. Then select cells B6F12 and PASTE. Now that we have some actual figures, we can refine our initial predictions in light
of the new information.
 Go back to the table on page 1 and fill in the column labeled “2nd predicted probability” with your
new, and hopefully improved, predictions.
(note: you may leave some the same, if you still agree with your initial prediction.)
Now that the table is complete for n=1-10 we will use the POWER OF TECHNOLOGY to apply
these ideas to groups with n = 10 to 365 members. To do this, COPY cells B5-F5 again. Then select
cells B6-F367 and PASTE. Your table should now be complete.
 Scroll down and look over the data for a moment.
 Go back to the table on page 1 and fill in the column labeled “actual probability.”
Were your initial predictions accurate?
Right on!
Pretty Close!
Not really.
Way off!
Were your second predictions accurate?
Right on!
Pretty Close!
Not really.
Way off!
Were they better than the initial predictions?
Much better!
A little better! Not really.
Describe your reaction to the actual probabilities? ________________________________________
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Can you identify any incorrect assumption(s) that you were making initially? In other words, explain
how you were thinking when you made your initial predictions. What have you since realized about
the problem that creates the striking results?
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