Set Theory: Probability

Laplace가 probability를 이렇게 정의하였다:

"The probability that A occurs" $:= \mathbb{P}[A] = \frac{|A|}{|S|} = \frac{|A|}{n}$ where $S$ is the sample space, $A$ is a subset of $S$, and $|S| = n$.

 

요즘은 중학교에서도 배우는 내용이겠지만 처음 정의됐을 때 만큼은 혁신적이었을 것이다. Probability theory는 도박류 게임을 연구하기 위해 1700~1800년대에는 많은 관심을 끌었다. 지금은 공학에서 필수적으로 배우는 학문 중 하나이다.

 

정의는 쉽게 이해가 되지만, Counting문제와 비슷하게 많이 까다로울 때가 많다.

 

From a standard deck of 52 cards, what is the probability of drawing (a hand of):

(a) three aces and two jacks?

(b) three aces and a pair?

(c) three of one kind and a pair (a full house)?

 

(a) Since we are getting a hand of cards, it is a set of cards (order does not matter, and without replacement).

We first choose 5 cards from 52 = 52C5. |S| = 52C5

There are 3 aces in total, so we have 4C3. |A| = 4C3

There are 4 jacks in total, so we have 4C2. |B| = 4C2

Pr(|A| intersection |B|) = |A| * |B| / |S| = (4C3)(4C2) / (52C5)

 

(b) A pair is any 2 cards of the same kind (value).

There are a total of 13 kinds of cards, but we already have 3 aces (one kind of 13), so |C| = (12C1) * (4C2)

Pr(|A| intersection |C|) = |A||C|/|S| = (4C3)(12C1)(4C2)/(52C5)

 

(c) Three of one kind is basically the 3x option of a pair.

There are again 13 kinds, so three of a kind and a pair (2 of a kind) requires 2 distinct kinds.

(13C1)(4C3) * (12C1)(4C2) / (52C5)

 

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