Introduction to Probability
The Birthday Paradox
Consider the question, how many people must be a room such that the probability of at least two will share a birthday exceeds 50%?
You might guess about 182 because it’s about half the days in a year, or 100 because it’s a nice round number. However, the answer might surprise you.
The correct answer is just 23 people. How can this be true?! The answer lies in probability theory. Let the event A be that of a group of k people not having any repeated birthdays, and let B be that of a group of k people having at least two who share a birthday, i.e. \(P(B) = 1 - P(A)\).
To calculate the number of k persons to exceed 50% one technique we might use is to decrement the number of possible shared birthdays for \(n+1\) people. For person 1 there is 100% chance they don’t share their birthday with another person, when adding person 2, there is a \(\frac{364}{365}\) chance they don’t share their birthday with person 1, and this continues as a conditional probability until you reach >50%.
\[P(A)= \frac{365}{365} \times \frac{364}{365} \times \cdots \times \frac{343}{365} \approx 0.493\] \[\therefore P(B) \approx 1 - 0.493 = 0.507\]A more concise way of representing this is the probability function \(\bar{p} = \frac{365!}{365^n(365-n)!}\). An approximation to try (that doesn’t require numerical methods) is \(n\geq \frac{1}{2} + \sqrt{\frac{1}{4} + 2\times\ln{2}\times 365}\).
A quick refresher on combinations and permutations. The number of combinations (order doesn’t matter) of n choose r is \(C(n, r)= \frac{n!}{r!(n-r)!}\) where \(!\) is the factorial operator (e.g. \(4! = 4\times 3\times 2\times 1 = 24\)). Similarly, the number of permutations (order does matter) is \(P(n, r) = \frac{n!}{(n-r)!}\). For both of these formulas, \(n\) is the number of total objects, and \(r\) is the number of objects chosen at once.
Conditional Probability
Conditional probability is the probablity of one thing happening given that another event has already occurred.
Joint Probability
Joint probability is the probability of two or more events occurring simultaneously.