Having just calculated this probability for two people, let us now calculate the probability of a 50% chance that three people (chosen from a group of "n" people) will have the same birthday. For calculating this probability, we will need to use this formula:
Where:
Filling in the right side of that formula with numbers:
c^{(k 1)} • k! = 365^{(3 1)} • 3! = 133,225 • 6 = 799,350
ln (1 / (1p)) = ln (1 / (1 .5)) = ln (1 / .5) = ln (2) = 0.6931471806
1 (N / c • (k + 1)) = 1 (88 / 365 • (3 + 1)) = 1 (88 / 1,460)) = 1 0.0602739726 = 0.9397260274
Then we multiply those 3 numbers 799,350 • 0.6931471806 • 0.9397260274 = 520,671.367621214
And its cube root equals 80.4491077898
Now, let's calculate the left side of the equation.
N • e^{(N / (c • k))} = 88 • 2.718281828459045^{(88 / 365 * 3)} = 88 • 2.718281828459045^{(0.0401826484)} = 88 • 0.9606139685 =
81.2045692937 which completes the left side calculation. Let us now calculate the "left side" and "right side" values when "N" = 86 and "N" = 87. (To save a lot of time and to minimize errors, the wisest way to make these calculations is to use an electronic spreadsheet such as Microsoft Excel™ or OpenOffice Calc™).
Now let's compare the left side and right side numbers as we vary the values of "N":
