Probability and Counting


Probability and Life

What is a sample space S?

The set of all possible outcomes of a (random) experiment.

What is an event?

A subset of the sample space.

What is probability?

  • The very naive definition of $P_{(A)}$
    • Number of favorable outcomes $A$ divided by all possible outcomes
    • Assumptions:
      • all outcomes are equally likely
      • the sample space is finite
  • The non-naive definition $P_{(A)}$ or Probability Axioms
    • A function which takes an input, event $A$, a subset of sample space $S$,
    • returns $P_{(A)} \in [0,1]$ as an output,
    • such that
      • the probability of the empty set $P_{\varnothing} = 0$
        • An event that is impossible to happy
      • the probability of the full set $P_{S} = 1$
        • Certainty: a guaranteed event that always happens
      • the probability of a countable infinite union equals to the sum of the probabilities of $A_1$,$A_2$,…, $A_n$ if they are disjoined (non-overlapping)
        • $P_{(\bigcup_{i=1}^{\infty} A_{n})} = \sum_{n=1}^{\infty}P_{A_n}$

What are some basic principles of counting?

  • Multiplication rules:

    • Independent events assuming
      • $P_{(A and B)}=P_{(A)}⋅P_{(B)}$
    • The general rule
      • $P_{(A and B)}=P_{(A)}⋅P_{(B|A)}$
      • If the events are independent, one happening doesn’t impact the probability of the other, and in that case $P_{(B|A)}=P_{(B)}$

  • Binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ = $\frac{n(n-1)(n-2)(n-k+1)}{k!}$

    • Example, what is the probability of having a full house card? (A full house has three cards of one kind and two of another)
    • First you choose a type of card (13 choices), then you choose three out of four of those cards, then you choose a second type of card, and finally you choose two of those four cards.
    • $P = \frac{\binom{13}{1} \cdot \binom{4}{3} \cdot \binom{12}{1 }\binom{4}{2}} {\binom{52}{5}} \approx 0.00144 $

How to choose $k$ objects out of $n$?

  • Sampling table
Order matters Order doesn’t matter
Without replacement S1: $n(n-1)…(n-k+1)$ S3: $\binom{n}{k}$
With replacement S2: $n^k$ S4:$\binom{n+k-1}{k}$
  • S1: The number of choices reduces every time
  • S2: $n$ choices each time
  • S3: $n$ choose $k$: think about how you choose the ice cream flavor and cones
  • S4:
    • Simple trivial cases
      • $k = 0$:
        • $P = \binom{n-1}{0} = 1$
        • not choosing is also a choice
      • $k = 1$:
        • $P = \binom{n}{1} = n$
        • you only choose once, there’s no difference whether there’s order or replacement
    • Simple non-trivial cases
      • $n = 2$:
        • $P = \binom{k+1}{k} = \binom{k+1}{1} = k + 1$
        • put $k$ particles in two boxes
    • General cases
      • how many ways are there to put $k$ indistinguishable particles in $n$ distinguishable boxes?
      • convert the question to a “dot and separator” code
      • $\binom{n+k-1}{k}$: $n+k-1$ positions, choose $k$ positions to put the dots
      • the same as $\binom{n+k-1}{n-1}$: $n+k-1$ positions, choose $n-1$ positions to put the seperators
      • e.g.
        • $n= 4$, $k=6$
        • $\cdot\cdot|\cdot|\cdot|\cdot\cdot$
        • 6 dots and 3 separators = 9 positions
        • choose 6 positions to put the dots
        • the same as choose 3 positions to put the seperators

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Terms

In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter (also called the fifth state of matter) which is typically formed when a gas of bosons at low densities is cooled to temperatures very close to absolute zero (-273.15 °C, -459.67 °F). Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which point microscopic quantum mechanical phenomena, particularly wavefunction interference, become apparent macroscopically. A BEC is formed by cooling a gas of extremely low density (about one-hundred-thousandth (1/100,000) the density of normal air) to ultra-low temperatures. This state was first predicted, generally, in 1924–1925 by Albert Einstein following and crediting a pioneering paper by Satyendra Nath Bose on the new field now known as quantum statistics.

In quantum statistics, Bose–Einstein (B–E) statistics describe one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete energy states at thermodynamic equilibrium.