Math 132A

Probability

Chance

• “What are the chances the Tigers will win this weekend?”
• “What’s the chance of rain tomorrow?”
• “What is the chance that a patient responds to a new therapy?”
• “What is the chance that an infant will pick the helper?”
• “What is the chance a coin will land heads up?”

Randomness

Random does not mean that everything is equally likely, or that there is no law or pattern at all!

Random does mean that we cannot predict the exact outcome for some reason.

• Truly random (quantum phenomena)

• Too complicated to predict

Theory of Probability

Way of constructing mathematical models of random phenomena.

What is a Model? (Cell biology)

Image by Science Primer (National Center for Biotechnology Information).

What is a Model? (Atoms)

What is a Model? (Planets)

Geocentric model:

{
  const context = DOM.context2d(500, 600);
  while (true) {
    const t = Date.now() / 1000;
    const x = 500/2 + 200*(Math.cos(t));
    const y = 500/2 + 200*(Math.sin(t));
    const xp = x + 60*Math.cos(t/10);
    const yp = y + 60*Math.sin(t/10);
    const xs = 500/2 + 200*(Math.cos(t + Math.PI/2));
    const ys = 500/2 + 200*(Math.sin(t + Math.PI/2));
    context.clearRect(0, 0, 500, 600);
    context.beginPath();
    context.fillStyle="green";
    context.arc(500/2, 500/2, 20, 0, 2 * Math.PI);
    context.fill();
    context.beginPath();
    context.fillStyle="orange";
    context.arc(xs, ys, 10, 0, 2 * Math.PI);
    context.fill();
    context.beginPath();
    context.fillStyle="red";
    context.arc(xp, yp, 10, 0, 2 * Math.PI);
    context.fill();
    context.beginPath();
    context.moveTo(500/2,500/2);
    context.lineTo(xs,ys);
    context.stroke();
    context.beginPath();
    context.moveTo(500/2,500/2);
    context.lineTo(x,y);
    context.lineTo(xp,yp);
    context.stroke();
    yield context.canvas;
  }
}

Heliocentric model:

{
  const context = DOM.context2d(500, 600);
  while (true) {
    const t = Date.now() / 1000;
    const xe = 500/2 + 100*(Math.cos(t/10));
    const ye = 500/2 + 200*(Math.sin(t/10));
    const xp = 500/2 + 130*(Math.cos(t/18 + Math.PI/3));
    const yp = 500/2 + 250*(Math.sin(t/18 + Math.PI/3));
    context.clearRect(0, 0, 500, 600);
    context.beginPath();
    context.fillStyle="yellow";
    context.arc(500/2, 1500/4, 20, 0, 2 * Math.PI);
    context.fill();
    context.beginPath();
    context.fillStyle="green";
    context.arc(xe, ye, 10, 0, 2 * Math.PI);
    context.fill();
    context.beginPath();
    context.fillStyle="red";
    context.arc(xp, yp, 10, 0, 2 * Math.PI);
    context.fill();
    yield context.canvas;
  }
}

What is a Model? (Geography)

What is a Model? (Free fall)

\[h(t) = h_0 + v_0 t - \frac{1}{2}gt^2\]

• \(h_0\): initial height
• \(v_0\): initial velocity
• \(g\): gravitational constant

What are models good for?

• To understand how the world works

• To predict, calculate or plan practical solutions

  Why?  

All models are wrong …

but some are useful.

George Box

Probability Models

• Experiment
• Outcome
• Event
• Probability

Random Experiments

A random experiment is an action or process that leads to one of several possible outcomes.

• flipping a coin and observing the side facing up

• measuring the air temperature at a given location at a given time

• organizing an election for an office

• testing 200 patients for Covid-19

These are the things that we are trying to model.

Outcomes

An outcome in an experiment is the observable result after conducting the experiment.

• One of the two faces on the coin.

• A number with unit, for example \(21^\circ\mathrm{C}\).

• The winner of the election.

• The list of results of the 200 tests performed.

The set of all possible outcomes is called the sample space of the experiment.

Events

An event is a collection of outcomes.

• A temperature between \(15^\circ\mathrm{C}\) and \(25^\circ\mathrm{C}\).

• A left-leaning candidate won.

• 30 out of the 200 patients tested positive.

Events can be referred to by letters.

• Suppose event \(A\) is the event of rolling a number smaller than 3 on a die. \[A = \{1, 2 \} \]

Probability

The probability of an event tells us how likely is that event to happen.

• Subjective probability: A number that quantify our belief that the event will occur.

• Frequency based probability: The proportion of times the event would occur if the random phenomenon could be observed an infinite number of times.

Example: flipping a single coin

Outcomes:

Head and Tail

Example: flipping a single coin

Probabilities:

• Both outcomes are equally likely

• Their probabilities have to add to 1.

• \(\operatorname{P}(\text{Head}) = \operatorname{P}(\text{Tail}) = \frac{1}{2}\)

Parameters

Mathematical models often have parameters.

\[h(t) = {\color{red} h_0} + {\color{green} v_0} t - \frac{1}{2}{\color{blue}g}t^2\]

• \(\color{red}h_0\): initial height
• \(\color{green}v_0\): initial velocity
• \(\color{blue}g\): gravitational constant

The probability of Head and the probability of Tail

The Law of Large Numbers

When repeating an experiment many times, the proportion of times a specific outcome is observed converges to the probability of that outcome.

Example: single coin again

viewof flip = Inputs.button("Flip")

import { aq, op } from '@uwdata/arquero'
data = aq.table({index: [...Array(1000).keys()]})
    .derive({heads: d => Math.random() < 0.5})
    .derive({ prop: aq.rolling(d => op.average(d.heads))})
head = FileAttachment("head.jpg")
tail = FileAttachment("tail.jpg")

done = data.slice(0,flip+1)
{if(done.get('heads', flip)) {return head.image()} else {return tail.image()}}

Plot.plot({
    marks: [
        Plot.ruleY([.5], {stroke: "red"}),
        Plot.lineY(done, {
            x: "index",
            y: "prop"
}),
]
})