Lecture 3: Birthday Problem, Properties of Probability

Birthday Problem

• In k people, let's find the probability that 2 have same birthdays.

  • Just for simplicity, exclude Feb 29th; assume other 265 days equally likely; assume independent of births.

• If k > 365, prob. is 1.

  • ex. | o o o | o o | ... | o o | -> Jan 1th, Jan 2th, ... Dec 31th (pigeonhole principle)

• Let k <= 365,

  $P(no\; match) = \frac{365\; * 364\; * 363\; ... \;(365-k+1)}{365^k}$

• $\begin{pmatrix}k\\2\\ \end{pmatrix} = \frac{k(k-1)}{2}$

• $\begin{pmatrix}23\\2\\ \end{pmatrix} = \frac{23 * 23}{2} = 253$

Axioms

• assume $0 <= P(A) <= 1$

1. $P(ϕ)=0, P(5)=1$

2. $P(\displaystyle⋃^{∞}_{n=1} An)=\displaystyle\sum^{k}_{j=0}P(An)$ if A1, A2, ... are disjoint (non-overlap)

   : It means that it's just sum of the prob!

• Properties

1. $P(A^c) = 1 - P(A)$

   • Proof

     : $1 = P(S) = P(A \cup A^c) = P(A) + P(A^c)$ since $A \cap A^c = ϕ$

2. If $A \subseteq B$, then $P(A) \subseteq P(B)$

   • Proof

     : $B = A \cup (B \cap A^c)$, disjoint $P(B) = P(A) + P(B \cap A^c) >= P(A)$

3. $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ disjointification

   • Proof

     : $P(A \cup B) = P(A \cup (B \cap A^c)) = P(A) + P(B \cap A^c)$

   • inclusion-exclusion 포함 배제의 원리

     : $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

     equiv to $P(A \cap B) + P(B \cap A^c) = P(B)$,

     True! since $A \cap B, A^c \cap B$ are disjoint, union is $B$.

   • $P(A \cup B \cup C)$

     = $P(A) + P(B) + P(C)$

     $- P(A \cap B) - P(A \cap C) - P(B \cap C)$

     $+ P(A \cap B \cap C)$

   • $P(A_1 \cup A_2 \cup\; ... \; \cup A_n)$

     $= \displaystyle \sum^{n}_{j=1}P(Aj) - \displaystyle \sum_{i < j}P(A_i \cap A_j) + \displaystyle \sum_{i < j < k}P(A_i \cap A_j \cap A_k) \; ...\;$

     $+(-1)^{n+1} \times P(A_1 \cap A_2 \cap \; ... \; \cap A_n)$

deMontmort's Problem

• In 1713, matching problem

  • 카드가 놓인 위치와 카드에 쓰인 숫자가 일치할 확률?

• n cardo, labeled 1, 2, ..., n. Let $A_j$ be the event "jth card matches"

  • $P(A_j) = \frac{1}{n}$, since all positions equally likely for card labeled j

  • $P(A_1 \cap A_2) = \frac{(n-2)!}{n!} = \frac{1}{n(n-1)} = \begin{pmatrix}n\\2\\ \end{pmatrix}$

  • $P(A_1 \cup A_2 \cup\; ... \; \cup A_n)$

    $= n \times \frac{1}{n} - \frac{(n-2)!}{n!} \times \frac{1}{n(n-1)} + \frac{n(n-1)(n-2)}{3} \times \frac{1}{n(n-1)(n-2)}-...$

    $= 1 - \frac{1}{2!} + \frac{1}{3!} - ... + (-1)^n \frac{1}{n!}$ -> Tayler's series

    $\approx 1 - \frac{1}{e}$

• 출처 : Statistics 110, boostcourse