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PHYS 260 Introductory Biophysics 3A. Probabilities 1

Introduction to probabilities. Useful for thermodynamics and diffusion parts of the course. Gambling: the driving force behind early research in probability. Blaise Pascal, Pierre Fermat

Statistical Physics Quantum Physics Biophysics (diffusion) Stock Market Genetics Engineering

Example: Acid-base behavior of Amino acids Amino Acids (AA)

Due to interactions with other AA molecules and with the solvent molecules some AA deprotonate at lower pH and others at higher pH values. Each AA behave somewhat different. pK

RH

-

R + H+

fraction protonated

1.0 1 pH unit

0.5

0.0

If 1 mole→ 6×10&' molecules

pH

One AA molecule is either protonated (RH) or deprotonated (R )

-

It is impossible to manually identify one molecule→ we should address collective behavior.

Symbols

One dice: 6 possible outcomes Two dice: 36 ordered pairs Rolling the dice: EXPERIMENT Each ordered pair is an OUTCOME Outcome is a simplest EVENT Events can also be combinations of several outcomes

EVENT: “rolling a seven”

EVENT: “…..” SAMPLE SPACE: all possible outcomes Venn Diagram

EXPERIMENT: two coin tosses Sample Space:

Event: At least one head occurs

NB: Experiments have to be REPEATABLE Sample space is a mathematical model of a real-life situation. One is supposed to be able to make some falsifiable predictions and to test them in experiments.

More definitions:

Sample Space

Union (“OR”) Intersection (“AND”) Complement

+ Empty Set

Discrete sample spaces: outcomes can be counted using positive integers (but there may be infinite number of them; Example: all possible sequences with head after many tails)

Compare: Non-discrete space example: all REAL numbers from 0 to 1. Introducing PROBABILITY: A number between 0 and 1 is assigned to each outcome, such that the sum of those numbers for all outcomes in S is equal to ONE. Intuitive probability definition: that number from 0 to 1 is proportional to a likelihood of the respective outcome. Dice: all numbers from 1 to 6 equally likely, so P(1)=…=P(6)=1/6 Uniform distribution. Probability of an event comprising several outcomes (“OR”) is sum of the probabilities of individual outcomes.

How many people should there be in a room for two of them having the same birthday (only the day and the month count) with a probability of ½ ? (r=?) 365 possible outcomes (ignoring leap years) for the experiment defined as “birthday of one isolated person”. Sample space is the set of all possible lists of r birthdays r>365 ® P=1 Number of distinct lists = 365r (may or may not have identical birthdays included) Assumption: all lists are equally likely (true in modern world, may not be true for some primitive agrarian societies – “harvest feast children”) Event A: “at least two people have same birthday”. Easier to calculate complimentary probability P(Ac), i.e. for all people having different birthdays.

P(A; r=23)=1/2

P(A; r=50)=0.97

6/54 Lotto 54x53x52x51x50x49 ways to fill ORDERED list with non-repeating numbers How do we take into account that the order does not matter? Divide by 6x5x4x3x2x1=6!=720 “Factorial”

Distinct sets with any ordering Total number of sets; probability of winning 1/25,827,165 In 6/49 Lotto: 1/13,983,816

Independence of events: the probability of one event (e.g. coin toss result) is independent of the previous events. Considering the tosses of generalized coin, P(H)=p, P(T)=q=1-p, p¹0.5 Bernoulli trials: “success” versus “failure” Event

According to product rule, P(Ai)=qi- 1p Event

(this are disjoint events, as one cannot get heads for the first time for two or more different i) = Geometric series, converges for p0 (the probability that life exists on a planet chosen at random; >0 because we exist) q...