Video to come!

There are three problems with the program we have set up thus far.

 

1. It is not always easy to identify the assumed “equally likely” components of an experiment.

 Consider, for example, simultaneously rolling a pair of dice and computing the sum of the two numbers you see. The sample space for this experiment is:

\(S=\{2,3,4,5,6,7,8,9,10,11,12\}\).

But people say that the individual outcomes here, \(\{2\}\), \(\{3\}\), \(\{4\}\),…, \(\{12\}\), are not equally likely.

Somehow we are meant to know that the underlying equally likely outcomes of this experiment are not actually the sums we see, but the pairs of numbers behind each sum with order of the pairs considered important. (This is actually confusing!)

There are possible ordered pairs and the set of all these pairs is, supposedly, the true sample space of the experiment, all deemed the equally likely.

1-1        1-2        1-3        1-4        1-5        1-6

2-1        2-2        2-3        2-4        2-5        2-6

3-1        3-2        3-3        3-4        3-5        3-6

4-1        4-2        4-3        4-4        4-5        4-6

5-1        5-2        5-3        5-4        5-5        5-6

6-1        6-2        6-3        6-4        6-5        6-6

Seeing that only one of these pairs gives a sum of 2 we now say that

\(p\left(\{2\}\right)=\dfrac{1}{36}\).

In general, we see

(Thus a sum of seven, for instance, is the most likely sum from rolling a pair of dice.)

Philosophical Question: We said that the order in which pairs of numbers presented is deemed important. Is this obvious? Shouldn’t rolling a six and a three be considered the same outcome as rolling a three and a six? In which case, do the values of all these probability computations change? (We resolve this question on lesson 1.6.)

This issue of identifying the “equally likely” components of an action can be considerably more difficult for more complicated experiments. How are we meant to know if the individual outcomes of an experiment we see are or aren’t the fundamental equally likely outcomes of that experiment?

Example: Lila has three children. There are four possibilities: She has three girls, she has two girls and a boy, she has two boys and a girl, or she has three boys. Are these the “equally likely” options?

Example: I could either win the lottery or lose the lottery.  There are just these two outcomes. Does this mean I have a 50% chance of winning the lottery?

And sometimes it is not even clear if the term “equally likely” is meaningful in some experiments.

Example: Consider the command: Pick a whole number at random. What does this mean? Is each number equally likely to be chosen? Does the term “equally likely” even apply? Think about it. Is someone likely to choose a thirteen-billion digit number? Among an infinitude of numbers, is it even theoretically possible that each has equal chance of being chosen?

Example: THE WALLET GAME

Two people decide to play the following game:

       Each pulls out her wallet.

       Whoever possesses the least amount of money in her wallet wins.

       Her prize? The contents of the other player’s wallet.

Each person can reason: “I stand to win more than I lose. Thus the game is in my favor.”

A game can’t be favorable simultaneously to both players! Something is very strange here. What are the odds of winning? Is everything really balanced and “equally likely”?

 2. Our definition of probability is philosophically flawed.

 Our definition of the probability of an event involves the phrase “equally likely.” And what does “equally likely” mean: it means equally probable. So using the term “equally likely” assumes you already know what probability is!

Our definition of probability is logically circular.

 

3. A natural extension of our work has some disturbing consequences.

We defined for an experiment with sample space \(S\) the probability of an event \(E\) occurring to be the value

\(p\left(E\right)=\dfrac{\text{size of }E}{\text{size of }S}\)

(under the assumption that each individual outcome is equally likely).

It is natural to extend this definition to a geometric setting where “size” is interpreted as length, area, or volume.

Example: A bus arrives at my local bus stop at the start of each and every hour, each and every day. The bus waits at the stop for precisely six minutes before taking off. I arrive at the bus stop at a random time of day and see no bus at the stop. What is the probability that I need wait no more than fifteen minutes for the next bus to arrive?

Answer: Here is a timeline showing the minutes at which the bus is waiting at the bus stop:

We are told that we arrive within a 56-minute period of this timeline and we are wondering what the chances are that we are actually within the final 15 minutes of that period? Here we have a sample space of size 54 minutes and we interested in an event of size 15 minutes. The probability we seek is thus:

\(p=\dfrac{15}{54}=\dfrac{5}{18}\).

Example: A circle sits snugly inside a square as shown. If a point inside the square is chosen at random, what at the chances that this point sits outside the circle?

Answer: Suppose the circle has radius \(r\) and the square has side length \(2r\). The shaded region is the region of interest (it represents our event \(E\)) and it has area \(\left(2r\right)^{2}-\pi r^2 = \left(4-\pi\right)r^2\).  The area of the entire square (our sample space \(S\) ) is \(\left(2r\right)^2=4r^2\). The probability we seek is thus

\(p\left(E\right)=\dfrac{\text{size of }E}{\text{size of }S}=\dfrac{\left(4-\pi\right)r^2}{4r^2}=\dfrac{4-\pi}{4}\).

Challenge: A sphere sits snugly inside a cube. A point inside the cube is chosen at random. Show that there the probability that the this point lands outside the sphere is \(\dfrac{8-\frac{4}{3}\pi}{8}=\dfrac{24-4\pi}{24}\) .

Philosophical Challenge:  What does “equally likely” mean in this geometric setting? Is the probability of picking any specific point zero? If so, is the probability of picking a point from a desired collection of points also zero? Could we land on a boundary point? (This issue is dealt with in the next lesson.)

And while we are mired in philosophical woes, consider the following disturbing problem:

Example: BERTRAND’S PARADOX

A chord is chosen at random in a circle. What is the probability that the chord is longer than the side length of an inscribed equilateral triangle?

Answer 1: By rotating the circle we may as well assume that one end of the chosen chord is positioned at the left end of the circle. Then we can see that the chord will be longer than the side of an inscribed equilateral triangle if its second end lies in the shaded portion of the circumference shown.

This represents \(\dfrac{1}{3}\) of the circumference of the circle. Thus the probability we seek is \(p = \dfrac{1}{3}\).

This argument is mathematically sound and the result is absolutely correct.

Answer 2: By rotating the circle we may as well assume that the chosen chord is horizontal. Then the chosen chord will be longer than the side-length of an inscribed equilateral triangle if its mid-point lies on the shaded portion of the diameter shown:

An exercise in geometry shows that this represents \(\dfrac{1}{2}\) of the diameter. Thus the probability we seek is \(p=\dfrac{1}{2}\).

This argument is mathematically sound and the result is also absolutely correct!

As French mathematician Joseph Bertrand (1822 – 1900) pointed out, the problem here is that the term “at random” is absolutely vague! The first answer defines “at random” to mean: select a point on the circumference of the circle and connect it with a given previously chosen point. The second solution assumes “at random” means: draw a circle on the floor and roll a broom handle from one side of the room across the circle.

It is possible to define “at random” a variety of different ways for this problem and arrive at different, but absolutely valid, answers. (If one draws a circle on a piece of paper and drops straws from above onto the figure, one finds that about \(\dfrac{1}{4}\) of them give chords of the length we seek! Here we want the midpoints of the chords the straws define to land inside a concentric circle of half the radius.)

Examples like these paradoxes alerted mathematicians to the problems with beginning approaches to probability theory. Terms such as “equally likely” and “at random” and even “probability” itself are fundamentally circular and vague notions.

No wonder one’s intuition is so challenged by this subject!

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