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4 changes: 2 additions & 2 deletions source/probability_theory_1a.Rmd
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Expand Up @@ -305,7 +305,7 @@ example of a *compound event* (see above) would be the event $W \text{and} R$,
meaning, the event where the Commanders won the game _and_ it rained.
:::

## Mutually exclusive events — the addition rule
## Mutually exclusive events — the addition rule {#sec-addition-rule}

**Definition:** If there are just two events $A$ and $B$ and they are "mutually
exclusive" or "disjoint," each implies the absence of the other. Green and red
Expand Down Expand Up @@ -855,7 +855,7 @@ $0.65 * 0.7 = 0.455$ = the probability of a nice day and a win. That is the
answer we seek. The method seems easy, but it also is easy to get confused and
obtain the wrong answer.

## Multiplication rule
## Multiplication rule {#sec-multiplication-rule}

We can generalize what we have just done. The foregoing formula exemplifies
what is known as the "multiplication rule":
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Expand Up @@ -85,18 +85,34 @@ cards.
A probability estimate of .2 indicates that you think there is twice as great
a chance of the event happening as if you had estimated a probability of .1.
This is the rock-bottom interpretation of the term "probability," and the
heart of the concept. [^expressing-p]

[^expressing-p]: A given probability may be expressed in
terms of probability, odds, or chances, and I shall use all
three terms to help familiarize you with them. If the
chances are 1 in 10, the odds are 9 to 1, and the
probability is .1. If the odds are 2 to 5, the chances are
5 in 7, and the probability is 5/7. If the odds are 99 to
1, the chances are 1 in 100, and the probability is .01. If
the odds are 100 to 1, the chances are 1 in 101, and the
probability is 1/101. "Likelihood" is a term related to
"probability" but is not a complete synonym for it.
heart of the concept.

:::{.callout-note}
## Expressing probability

A given probability may be expressed in terms of probability, odds, or chances,
and I shall use all three terms to help familiarize you with them.

Let us say we think there is a probability of 0.1 that it will rain tomorrow.

We can restate this probability by saying there is a one in 10 *chance* that it
will rain tomorrow ($1 / 10 = 0.1$). Giving the *chances* as 1 in 10, or 2 in
20, or 10 in 100, is the same as saying the probability is 0.1.

If we multiply the probability by 100 we get the *percent chance* — another way
of saying the probability. Here we have a $0.1 * 100$ = 10% chance of rain. We
could also say that the chances of rain are 10 in 100.

*Odds* are still another way of expressing probability. Here we think of our
outcome of interest — a day *with rain* and compare it to our outcome that is
not of interest — a day *without rain*. Our probability of 0.1 means that we
expect one day *with rain* in every 10 days, and therefore, one day *with rain*
for every nine days *without rain*. We can express the 0.1 probability of rain
as *odds* 1 to 9 (of a rainy day), or 9 to 1 *against* a rainy day.

"Likelihood" is a term related to "probability" but is not a complete synonym
for it — it has a specific and technical meaning in probability and statistics.
:::

The idea of probability arises when you are not sure about what will happen in
an uncertain situation. For example, you may lack information and therefore
Expand Down Expand Up @@ -183,7 +199,7 @@ There are two major concepts and points of view about probability —
in others. Though they may seem incompatible in principle, there almost never
is confusion about which is appropriate in a given situation.

1. *Frequency*. The probability of an event can be said to be the
1. *Frequency*: The probability of an event can be said to be the
proportion of times that the event has taken place in the past,
usually based on a long series of trials. Insurance companies use
this when they estimate the probability that a thirty-five-year-old
Expand All @@ -193,37 +209,35 @@ is confusion about which is appropriate in a given situation.
and so you cannot reasonably reckon the proportion of times they
occurred one way or the other in the past.)

2. *Degree of belief*. The probability that an event will take place or that
2. *Degree of belief*: The probability that an event will take place or that
a statement is true can be said to correspond to the odds at which you
would bet that the event will take place. (Notice a shortcoming of this
concept: You might be willing to accept a five-dollar bet at 2-1 odds that
your team will win the game, but you might be unwilling to bet a hundred
dollars at the same odds.)

See [@barnett1982comparative, chapter 3] for an in-depth
discussion of different approaches to probability.
See [@barnett1982comparative, chapter 3] for an in-depth discussion of
different approaches to probability.

The connection between gambling and immorality or vice troubles some people
about gambling examples. On the other hand,
the immediacy and consequences of the decisions that the
gambler has to make give the subject a special tang. There
are several reasons why statistics use so many gambling
examples — and especially tossing coins, throwing dice, and
playing cards:

1. *Historical*. The theory of probability began with gambling
about gambling examples. On the other hand, the immediacy and consequences of
the decisions that the gambler has to make give the subject a special tang.
There are several reasons why statistics use so many gambling examples — and
especially tossing coins, throwing dice, and playing cards:

1. *Historical*: The theory of probability began with gambling
examples of dice analyzed by Cardano, Galileo, and then by Pascal
and Fermat.
2. *Generality*. These examples are not related to any particular walk
2. *Generality*: These examples are not related to any particular walk
of life, and therefore they can be generalized to applications in
any walk of life. Students in any field — business, medicine,
science — can feel equally at home with gambling examples.
3. *Sharpness*. These examples are particularly stark, and
3. *Sharpness*: These examples are particularly stark, and
unencumbered by the baggage of particular walks of life or special
uses.
4. *Universality*. Many other texts use these same examples, and therefore
the use of them connects up this book with the main body of writing about
probability and statistics.
4. *Universality*: Many other texts use these same examples, and therefore the
use of them connects up this book with the main body of writing about
probability and statistics.

Often we'll begin with a gambling example and then consider an example
in one of the professional fields — such as business and other
Expand All @@ -237,11 +251,11 @@ any interest in the use of their work in public policy.

## Back to Proxies

Example of a proxy: The "probability risk assessments" (PRAs) that are
made for the chances of failures of nuclear power plants are based, not
on long experience or even on laboratory experiment, but rather on
theorizing of various kinds — using pieces of prior experience wherever
possible, of course. A PRA can cost a nuclear facility \$5 million.
Example of a proxy: The "probability risk assessments" (PRAs) that are made for
the chances of failures of nuclear power plants are based, not on long
experience or even on laboratory experiment, but rather on theorizing of
various kinds — using pieces of prior experience wherever possible, of course.
A PRA can cost a nuclear facility many millions of dollars.

Another example: If a manager of a high-street store looks at the sales of a
particular brand of smart watches in the last two Decembers, and on that basis
Expand Down Expand Up @@ -281,18 +295,18 @@ odd). Here are several general methods of estimation, where we define each metho
watch long enough you might come to estimate something like 6 in
52.)

General information and experience are also the source for
estimating the probability that the sales of a particular brand of smart
watch this December will be between 200 and 250, based on sales the last
two Decembers; that your team will win the football game tomorrow; that
war will break out next year; or that a United States astronaut will reach
Mars before a Russian astronaut. You simply put together all your relevant
prior experience and knowledge, and then make an educated guess.
General information and experience are also the source for estimating the
probability that the sales of a particular brand of smart watch this
December will be between 200 and 250, based on sales the last two
Decembers; that your team will win the football game tomorrow; that war
will break out next year; or that a United States astronaut will reach Mars
before a Chinese astronaut. You simply put together all your relevant prior
experience and knowledge, and then make an educated guess.

Observation of repeated events can help you estimate the probability
that a machine will turn out a defective part or that a child can
memorize four nonsense syllables correctly in one attempt. You watch
repeated trials of similar events and record the results.
Observation of repeated events can help you estimate the probability that
a machine will turn out a defective part or that a child can memorize four
nonsense syllables correctly in one attempt. You watch repeated trials of
similar events and record the results.

Data on the mortality rates for people of various ages in a
particular country in a given decade are the basis for estimating
Expand Down Expand Up @@ -333,11 +347,11 @@ odd). Here are several general methods of estimation, where we define each metho
the probability of a shuttle failure. (Probabilists have made some rather
peculiar attempts over the centuries to estimate probabilities from the
length of a zero-defect time series — such as the fact that the sun has
never failed to rise (foggy days aside! — based on the undeniable fact
that the longer such a series is, the smaller the probability of a
failure; see e.g., [@whitworth1897dcc, pp. xix-xli]. However, one surely
has more information on which to act when one has a long series of
observations of the same magnitude rather than a short series).
never failed to rise (foggy days aside!) — based on the undeniable fact
that the longer such a series is, the smaller the probability of a failure;
see e.g., [@whitworth1897dcc, pp. xix-xli]. However, one surely has more
information on which to act when one has a long series of observations of
the same magnitude rather than a short series).

2. **Simulated experience.**

Expand All @@ -355,7 +369,7 @@ odd). Here are several general methods of estimation, where we define each metho
even-numbered spades in the deck of fifty-two cards — using the *sample
space analysis* you see below. No reason at all. But that procedure would
not work if you wanted to estimate the probability of a baseball batter
getting a hit or a cigarette lighter producing flame.
getting a hit or a lighter producing flame.

Some varieties of poker are so complex that experiment is the only
feasible way to estimate the probabilities a player needs to know.
Expand Down Expand Up @@ -386,24 +400,25 @@ odd). Here are several general methods of estimation, where we define each metho

4. **Mathematical shortcuts to sample-space analysis.**

A fourth source of probability estimates is *mathematical
calculations*. If one knows by other means that the probability of
a spade is 1/4 and the probability of an even-numbered card is 6/13,
one can use probability calculation rules to calculate that the
probability of turning up an even-numbered spade is 6/52 (that is, 1/4 x
6/13). If one knows that the probability of a spade is 1/4 and the
probability of a heart is 1/4, one can then calculate that the probability
of getting a heart *or* a spade is 1/2 (that is 1/4 + 1/4). The point here
is not the particular calculation procedures, which we will touch on
later, but rather that one can often calculate the desired probability on
the basis of already-known probabilities.

It is possible to estimate probabilities with mathematical
calculation only if one knows *by other means* the probabilities of
some related events. For example, there is no possible way of
mathematically calculating that a child will memorize four nonsense
syllables correctly in one attempt; empirical knowledge is
necessary.
A fourth source of probability estimates is *mathematical calculations*.
(We will introduce some probability calculation rules in
@sec-prob-theory-one-b.) If one knows by other means that the probability
of a spade is 1/4 and the probability of an even-numbered card is 6/13, one
can use probability calculation rules to calculate that the probability of
turning up an even-numbered spade is 6/52 (that is, 1/4 x 6/13). (This is *multiplication rule* introduced in @sec-multiplication-rule). If one
knows that the probability of a spade is 1/4 and the probability of a heart
is 1/4, one can then calculate that the probability of getting a heart *or*
a spade is 1/2 (that is 1/4 + 1/4). (We are using the *addition rule* from
@sec-addition-rule.) The point here is not the particular calculation
procedures, which we will touch on later, but rather that one can often
calculate the desired probability on the basis of already-known
probabilities.

It is possible to estimate probabilities with mathematical calculation only
if one knows *by other means* the probabilities of some related events. For
example, there is no possible way of mathematically calculating that
a child will memorize four nonsense syllables correctly in one attempt;
empirical knowledge is necessary.

5. **Kitchen-sink methods.**

Expand Down Expand Up @@ -759,7 +774,7 @@ determinacy-indeterminacy and predictable-unpredictable. What matters for
*decision purposes* is whether you can predict. Whether the process is
"really" determinate is largely a matter of definition and labeling, an
unnecessary philosophical controversy for our purposes (and
perhaps for any other purpose) [^quanta-decisions].
perhaps for any other purpose).[^quanta-decisions]

[^quanta-decisions]: The idea that our aim is to advance our work in improving
our knowledge and our decisions, rather than to answer "ultimate" questions
Expand Down Expand Up @@ -794,7 +809,7 @@ probabilities you work with are influenced by your knowledge of the
facts of the situation.

Admittedly, this way of thinking about probability takes some getting used to.
Events may appear to be random, but in fact, we can predict them — and *visa
Events may appear to be random, but in fact, we can predict them — and *vice
versa*. For example, suppose a magician does a simple trick with dice such as
this one:

Expand Down Expand Up @@ -862,7 +877,21 @@ independent of each other is important in statistical inference. For example,
as we will see later, when we add many unpredictable deviations together, and
plot the distribution of the result, we end up with the famous and very common
bell-shaped *normal distribution* — this striking result comes about because
of a mathematical phenomenon called the Central Limit Theorem. We will show this at work, later in the book.
of a mathematical phenomenon called the Central Limit Theorem.[^clt]

<!---
We may show this at work, later in the book.
It does come up in the confidence_2 chapter, but without further explanation.
-->

[^clt]: The Central Limit Theorem is an interesting mathematical result that
proves something you can show for yourself by simulation — that if we take
means of many values drawn from *any* shape of distribution, and then look at
the distribution of the resulting means, it will be close to the *normal*
(bell-curve) distribution. If you are interested in a technical
(mathematical) explanation of this result, see [the Wikipedia page on the
Central Limit Theorem](https://en.wikipedia.org/wiki/Central_limit_theorem).

## Randomness from the computer

Expand Down Expand Up @@ -1004,13 +1033,15 @@ well as to (2) and (4), to bring out the important fact that the
procedure is the same as in resampling questions in statistics.

One could easily produce examples like (1) and (2) for cases that are
similar except that the drawing is without replacement, as in the
sampling version of Fisher's permutation test — for example, a tea
taster [@fisher1935design; @fisher1960design, Chapter II, section 5]. And one
could adduce the example of prices in different state liquor control systems
(see @sec-public-liquor) which is similar to cases (3) and (4) except that
sampling without replacement seems appropriate. Again, the analogs to cases (2)
and (4) would generally be called "resampling."
similar except that the drawing is without replacement.[^fishers-tea]
And one could adduce the example of prices in different state liquor control
systems (see @sec-public-liquor) which is similar to cases (3) and (4) except
that sampling without replacement seems appropriate. Again, the analogs to
cases (2) and (4) would generally be called "resampling."

[^fishers-tea]: One example of drawing *without replacement* is the sampling
version of Ronald Fisher's permutation test — see [@fisher1935design;
@fisher1960design, Chapter II, section 5].

The concept of resampling is defined in a more precise way in
@sec-what-is-resampling.
Expand All @@ -1034,7 +1065,8 @@ principle* that these processes can be "understood," certainly one can
develop a machine (or a baton twirler) that will make the outcome
predictable for many turns. But this has nothing to do with whether the
mechanism is "really" something one wants to say is influenced by
"chance." This is the point of the cooking-TV demonstration. The outcome traverses from non-chance (determined) to chance
(not determined) in a smooth way even though the physical mechanism that
"chance." This is the point of the demonstration with the sofa and TV ends of
the remote control. The outcome traverses from non-chance (determined) to
chance (not determined) in a smooth way even though the physical mechanism that
produces the revolutions remains much the same over the traverse.

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